Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- -5*c^3-2*d^2-3*c*d-3*c*d+3*c^3+10*d^2 если c=2
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- log(sqrt(10))*(6+x-x^2) если x=-4
- cos(2*a)-cos(a)^2 если a=-1/2
- Разложить многочлен на множители:
- 42*a*m+16*m*u-12*a*u-56*m^2
- z^3+8
- a^6+b^6
- 81*q^2-100*p^2
- Общий знаменатель:
- 2/(9*p-12*q)-4/(9*p+12*q)+4*p/(16*q^2-9*p^2)
- (t^2-81)/(6*t^2+1)*((54*t+1)/(t-9)+(54*t-1)/(t+9))
- (5*m-n)/(14*m)-m-6*n/7*m
- 1/b-1/a
- Умножение столбиком:
- 26*35
- 15*15
- 38*42
- 16*38
- Деление столбиком:
- 62/2
- 122/6
- 183/9
- 183/4
- Раскрыть скобки в:
- 4*(m+1)-(2*m+1)^2
- (6*x+1)*(6*x-1)
- (b+7)*(b-4)+(2*b-6)*(2*b+6)
- (a^3-b^4)*(a^3+b^4)
- Разложение числа на простые множители:
- 2024
- 300
- 20736
- 1260
- Производная:
- -1/3
- Интеграл d{x}:
-
-1/3
-
Идентичные выражения
- (sqrt((один +sin(a))/(один -sin(a)))-sqrt((один -sin(a))/(один +sin(a))))*cot(a) если a=- один / три
- ( квадратный корень из ((1 плюс синус от (a)) делить на (1 минус синус от (a))) минус квадратный корень из ((1 минус синус от (a)) делить на (1 плюс синус от (a)))) умножить на котангенс от (a) если a равно минус 1 делить на 3
- ( квадратный корень из ((один плюс синус от (a)) делить на (один минус синус от (a))) минус квадратный корень из ((один минус синус от (a)) делить на (один плюс синус от (a)))) умножить на котангенс от (a) если a равно минус один делить на три
- (√((1+sin(a))/(1-sin(a)))-√((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))cot(a) если a=-1/3
- sqrt1+sina/1-sina-sqrt1-sina/1+sinacota если a=-1/3
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) при a=-1/3
- (sqrt((1+sin(a)) разделить на (1-sin(a)))-sqrt((1-sin(a)) разделить на (1+sin(a))))*cot(a) если a=-1 разделить на 3
-
Похожие выражения
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1+sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- (sqrt((1+sin(a))/(1-sin(a)))+sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- (sqrt((1+sin(a))/(1+sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=+1/3
- (sqrt((1-sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
- (sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1-sin(a))))*cot(a) если a=-1/3
(sqrt((1+sin(a))/(1-sin(a)))-sqrt((1-sin(a))/(1+sin(a))))*cot(a) если a=-1/3
Решение
Вы ввели
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/ ____________ ____________\ | / 1 + sin(a) / 1 - sin(a) | | / ---------- - / ---------- |*cot(a) \\/ 1 - sin(a) \/ 1 + sin(a) /
$$\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \cot{\left(a \right)}$$
(sqrt((1 + sin(a))/(1 - sin(a))) - sqrt((1 - sin(a))/(1 + sin(a))))*cot(a)
Общее упрощение
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/ ________________ ____________\ | / -(1 + sin(a)) / 1 - sin(a) | | / -------------- - / ---------- |*cot(a) \\/ -1 + sin(a) \/ 1 + sin(a) /
$$\left(- \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}} + \sqrt{- \frac{\sin{\left(a \right)} + 1}{\sin{\left(a \right)} - 1}}\right) \cot{\left(a \right)}$$
(sqrt(-(1 + sin(a))/(-1 + sin(a))) - sqrt((1 - sin(a))/(1 + sin(a))))*cot(a)
Подстановка условия
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(sqrt((1 + sin(a))/(1 - sin(a))) - sqrt((1 - sin(a))/(1 + sin(a))))*cot(a) при a = -1/3
подставляем
/ ____________ ____________\ | / 1 + sin(a) / 1 - sin(a) | | / ---------- - / ---------- |*cot(a) \\/ 1 - sin(a) \/ 1 + sin(a) /
$$\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \cot{\left(a \right)}$$
/ ________________ ____________\ | / -(1 + sin(a)) / 1 - sin(a) | | / -------------- - / ---------- |*cot(a) \\/ -1 + sin(a) \/ 1 + sin(a) /
$$\left(- \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}} + \sqrt{- \frac{\sin{\left(a \right)} + 1}{\sin{\left(a \right)} - 1}}\right) \cot{\left(a \right)}$$
переменные
a = -1/3
$$a = - \frac{1}{3}$$
/ _____________________ _________________\ | / -(1 + sin((-1/3))) / 1 - sin((-1/3)) | | / ------------------- - / --------------- |*cot((-1/3)) \\/ -1 + sin((-1/3)) \/ 1 + sin((-1/3)) /
$$\left(- \sqrt{\frac{- \sin{\left((-1/3) \right)} + 1}{\sin{\left((-1/3) \right)} + 1}} + \sqrt{- \frac{\sin{\left((-1/3) \right)} + 1}{\sin{\left((-1/3) \right)} - 1}}\right) \cot{\left((-1/3) \right)}$$
/ ___________________ _______________\ | / -(1 + sin(-1/3)) / 1 - sin(-1/3) | | / ----------------- - / ------------- |*cot(-1/3) \\/ -1 + sin(-1/3) \/ 1 + sin(-1/3) /
$$\left(- \sqrt{\frac{- \sin{\left(- \frac{1}{3} \right)} + 1}{\sin{\left(- \frac{1}{3} \right)} + 1}} + \sqrt{- \frac{\sin{\left(- \frac{1}{3} \right)} + 1}{-1 + \sin{\left(- \frac{1}{3} \right)}}}\right) \cot{\left(- \frac{1}{3} \right)}$$
/ _______________ ______________\ | / -1 ______________ \/ 1 + sin(1/3) | -| / ------------- *\/ 1 - sin(1/3) - ----------------|*cot(1/3) |\/ -1 - sin(1/3) ______________| \ \/ 1 - sin(1/3) /
$$- \left(- \frac{\sqrt{\sin{\left(\frac{1}{3} \right)} + 1}}{\sqrt{- \sin{\left(\frac{1}{3} \right)} + 1}} + \sqrt{- \frac{1}{-1 - \sin{\left(\frac{1}{3} \right)}}} \sqrt{- \sin{\left(\frac{1}{3} \right)} + 1}\right) \cot{\left(\frac{1}{3} \right)}$$
-(sqrt(-1/(-1 - sin(1/3)))*sqrt(1 - sin(1/3)) - sqrt(1 + sin(1/3))/sqrt(1 - sin(1/3)))*cot(1/3)
Собрать выражение
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2*cot(a)*sin(a) ----------------------------- ____________ ____________ \/ 1 - sin(a) *\/ 1 + sin(a)
$$\frac{2 \sin{\left(a \right)} \cot{\left(a \right)}}{\sqrt{- \sin{\left(a \right)} + 1} \sqrt{\sin{\left(a \right)} + 1}}$$
2*cot(a)*sin(a)/(sqrt(1 - sin(a))*sqrt(1 + sin(a)))
Рациональный знаменатель
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_________________________ _________________________ / 1 sin(a) / 1 sin(a) / ---------- + ---------- *cot(a) - / ---------- - ---------- *cot(a) \/ 1 - sin(a) 1 - sin(a) \/ 1 + sin(a) 1 + sin(a)
$$\sqrt{\frac{\sin{\left(a \right)}}{- \sin{\left(a \right)} + 1} + \frac{1}{- \sin{\left(a \right)} + 1}} \cot{\left(a \right)} - \sqrt{- \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} + 1} + \frac{1}{\sin{\left(a \right)} + 1}} \cot{\left(a \right)}$$
sqrt(1/(1 - sin(a)) + sin(a)/(1 - sin(a)))*cot(a) - sqrt(1/(1 + sin(a)) - sin(a)/(1 + sin(a)))*cot(a)
Численный ответ
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(((1.0 + sin(a))/(1.0 - sin(a)))^0.5 - ((1.0 - sin(a))/(1.0 + sin(a)))^0.5)*cot(a)
(((1.0 + sin(a))/(1.0 - sin(a)))^0.5 - ((1.0 - sin(a))/(1.0 + sin(a)))^0.5)*cot(a)
Общий знаменатель
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_____________________________ _________________________ / 1 sin(a) / 1 sin(a) / - ----------- - ----------- *cot(a) - / ---------- - ---------- *cot(a) \/ -1 + sin(a) -1 + sin(a) \/ 1 + sin(a) 1 + sin(a)
$$\sqrt{- \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} - \frac{1}{\sin{\left(a \right)} - 1}} \cot{\left(a \right)} - \sqrt{- \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} + 1} + \frac{1}{\sin{\left(a \right)} + 1}} \cot{\left(a \right)}$$
sqrt(-1/(-1 + sin(a)) - sin(a)/(-1 + sin(a)))*cot(a) - sqrt(1/(1 + sin(a)) - sin(a)/(1 + sin(a)))*cot(a)
Степени
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/ ________________________ ________________________\ | / / -I*a I*a\ / / -I*a I*a\ | | / I*\- e + e / / I*\- e + e / | | / 1 - ------------------ / 1 + ------------------ | | / 2 / 2 | | / ---------------------- - / ---------------------- |*cot(a) | / / -I*a I*a\ / / -I*a I*a\ | | / I*\- e + e / / I*\- e + e / | | / 1 + ------------------ / 1 - ------------------ | \\/ 2 \/ 2 /
$$\left(- \sqrt{\frac{\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1}{- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1}} + \sqrt{\frac{- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1}{\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1}}\right) \cot{\left(a \right)}$$
(sqrt((1 - i*(-exp(-i*a) + exp(i*a))/2)/(1 + i*(-exp(-i*a) + exp(i*a))/2)) - sqrt((1 + i*(-exp(-i*a) + exp(i*a))/2)/(1 - i*(-exp(-i*a) + exp(i*a))/2)))*cot(a)
Комбинаторика
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/ _________________________ _________________________\ | / 1 sin(a) / 1 sin(a) | -| / ---------- - ---------- - / ---------- + ---------- |*cot(a) \\/ 1 + sin(a) 1 + sin(a) \/ 1 - sin(a) 1 - sin(a) /
$$- \left(- \sqrt{\frac{\sin{\left(a \right)}}{- \sin{\left(a \right)} + 1} + \frac{1}{- \sin{\left(a \right)} + 1}} + \sqrt{- \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} + 1} + \frac{1}{\sin{\left(a \right)} + 1}}\right) \cot{\left(a \right)}$$
-(sqrt(1/(1 + sin(a)) - sin(a)/(1 + sin(a))) - sqrt(1/(1 - sin(a)) + sin(a)/(1 - sin(a))))*cot(a)
Тригонометрическая часть
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____________ ____________
/ 1 + sin(a) / 1 - sin(a)
/ ---------- - / ----------
\/ 1 - sin(a) \/ 1 + sin(a)
-----------------------------------
tan(a)
$$\frac{\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}}{\tan{\left(a \right)}}$$
/ ____________ ____________\
| / 1 + sin(a) / 1 - sin(a) |
| / ---------- - / ---------- |*cos(a)
\\/ 1 - sin(a) \/ 1 + sin(a) /
--------------------------------------------
sin(a)
$$\frac{\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \cos{\left(a \right)}}{\sin{\left(a \right)}}$$
/ ____________ ____________\
| / 1 + sin(a) / 1 - sin(a) |
| / ---------- - / ---------- |*sin(2*a)
\\/ 1 - sin(a) \/ 1 + sin(a) /
----------------------------------------------
2
2*sin (a)
$$\frac{\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
/ ____________ ____________\
| / 1 + sin(a) / 1 - sin(a) | / pi\
| / ---------- - / ---------- |*sin|a + --|
\\/ 1 - sin(a) \/ 1 + sin(a) / \ 2 /
-------------------------------------------------
sin(a)
$$\frac{\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}}$$
/ ____________ ____________\
| / 1 / 1 |
| / 1 + ------ / 1 - ------ |
| / csc(a) / csc(a) |
| / ---------- - / ---------- |*csc(a)
| / 1 / 1 |
| / 1 - ------ / 1 + ------ |
\\/ csc(a) \/ csc(a) /
----------------------------------------------------
sec(a)
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\csc{\left(a \right)}}}{1 - \frac{1}{\csc{\left(a \right)}}}} - \sqrt{\frac{1 - \frac{1}{\csc{\left(a \right)}}}{1 + \frac{1}{\csc{\left(a \right)}}}}\right) \csc{\left(a \right)}}{\sec{\left(a \right)}}$$
/ ____________ ____________\
| / 1 / 1 |
| / 1 + ------ / 1 - ------ |
2 | / csc(a) / csc(a) |
csc (a)*| / ---------- - / ---------- |
| / 1 / 1 |
| / 1 - ------ / 1 + ------ |
\\/ csc(a) \/ csc(a) /
-----------------------------------------------------
2*csc(2*a)
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\csc{\left(a \right)}}}{1 - \frac{1}{\csc{\left(a \right)}}}} - \sqrt{\frac{1 - \frac{1}{\csc{\left(a \right)}}}{1 + \frac{1}{\csc{\left(a \right)}}}}\right) \csc^{2}{\left(a \right)}}{2 \csc{\left(2 a \right)}}$$
/ ____________ ____________\
/ 2/a\\ | / 1 + sin(a) / 1 - sin(a) |
|1 - tan |-||*| / ---------- - / ---------- |
\ \2// \\/ 1 - sin(a) \/ 1 + sin(a) /
---------------------------------------------------
/a\
2*tan|-|
\2/
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right)}{2 \tan{\left(\frac{a}{2} \right)}}$$
/ ____________ ____________\
| / 1 / 1 |
| / 1 + ------ / 1 - ------ |
| / csc(a) / csc(a) |
| / ---------- - / ---------- |*csc(a)
| / 1 / 1 |
| / 1 - ------ / 1 + ------ |
\\/ csc(a) \/ csc(a) /
----------------------------------------------------
/pi \
csc|-- - a|
\2 /
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\csc{\left(a \right)}}}{1 - \frac{1}{\csc{\left(a \right)}}}} - \sqrt{\frac{1 - \frac{1}{\csc{\left(a \right)}}}{1 + \frac{1}{\csc{\left(a \right)}}}}\right) \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ _________________ _________________\
| / / pi\ / / pi\ |
| / 1 + cos|a - --| / 1 - cos|a - --| |
| / \ 2 / / \ 2 / |
| / --------------- - / --------------- |*cos(a)
| / / pi\ / / pi\ |
| / 1 - cos|a - --| / 1 + cos|a - --| |
\\/ \ 2 / \/ \ 2 / /
--------------------------------------------------------------
/ pi\
cos|a - --|
\ 2 /
$$\frac{\left(\sqrt{\frac{\cos{\left(a - \frac{\pi}{2} \right)} + 1}{- \cos{\left(a - \frac{\pi}{2} \right)} + 1}} - \sqrt{\frac{- \cos{\left(a - \frac{\pi}{2} \right)} + 1}{\cos{\left(a - \frac{\pi}{2} \right)} + 1}}\right) \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ _________________ _________________\
| / 1 / 1 |
| / 1 + ----------- / 1 - ----------- |
| / csc(pi - a) / csc(pi - a) |
| / --------------- - / --------------- |*csc(pi - a)
| / 1 / 1 |
| / 1 - ----------- / 1 + ----------- |
\\/ csc(pi - a) \/ csc(pi - a) /
-------------------------------------------------------------------
/pi \
csc|-- - a|
\2 /
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\csc{\left(- a + \pi \right)}}}{1 - \frac{1}{\csc{\left(- a + \pi \right)}}}} - \sqrt{\frac{1 - \frac{1}{\csc{\left(- a + \pi \right)}}}{1 + \frac{1}{\csc{\left(- a + \pi \right)}}}}\right) \csc{\left(- a + \pi \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ _________________ _________________\
| / / pi\ / / pi\ |
| / 1 + cos|a - --| / 1 - cos|a - --| |
| / \ 2 / / \ 2 / | / pi\
| / --------------- - / --------------- |*cos|2*a - --|
| / / pi\ / / pi\ | \ 2 /
| / 1 - cos|a - --| / 1 + cos|a - --| |
\\/ \ 2 / \/ \ 2 / /
---------------------------------------------------------------------
2/ pi\
2*cos |a - --|
\ 2 /
$$\frac{\left(\sqrt{\frac{\cos{\left(a - \frac{\pi}{2} \right)} + 1}{- \cos{\left(a - \frac{\pi}{2} \right)} + 1}} - \sqrt{\frac{- \cos{\left(a - \frac{\pi}{2} \right)} + 1}{\cos{\left(a - \frac{\pi}{2} \right)} + 1}}\right) \cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos^{2}{\left(a - \frac{\pi}{2} \right)}}$$
/ _________________ _________________\
| / 1 / 1 |
| / 1 + ----------- / 1 - ----------- |
| / / pi\ / / pi\ |
| / sec|a - --| / sec|a - --| |
| / \ 2 / / \ 2 / | / pi\
| / --------------- - / --------------- |*sec|a - --|
| / 1 / 1 | \ 2 /
| / 1 - ----------- / 1 + ----------- |
| / / pi\ / / pi\ |
| / sec|a - --| / sec|a - --| |
\\/ \ 2 / \/ \ 2 / /
---------------------------------------------------------------------------
sec(a)
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}} - \sqrt{\frac{1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
/ _________________ _________________\
| / 1 / 1 |
| / 1 + ----------- / 1 - ----------- |
| / /pi \ / /pi \ |
| / sec|-- - a| / sec|-- - a| |
| / \2 / / \2 / | /pi \
| / --------------- - / --------------- |*sec|-- - a|
| / 1 / 1 | \2 /
| / 1 - ----------- / 1 + ----------- |
| / /pi \ / /pi \ |
| / sec|-- - a| / sec|-- - a| |
\\/ \2 / \/ \2 / /
---------------------------------------------------------------------------
sec(a)
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}}}{1 - \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}}}} - \sqrt{\frac{1 - \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}}}}\right) \sec{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
/ _________________ _________________\
| / 1 / 1 |
| / 1 + ----------- / 1 - ----------- |
| / / pi\ / / pi\ |
| / sec|a - --| / sec|a - --| |
2/ pi\ | / \ 2 / / \ 2 / |
sec |a - --|*| / --------------- - / --------------- |
\ 2 / | / 1 / 1 |
| / 1 - ----------- / 1 + ----------- |
| / / pi\ / / pi\ |
| / sec|a - --| / sec|a - --| |
\\/ \ 2 / \/ \ 2 / /
----------------------------------------------------------------------------
/ pi\
2*sec|2*a - --|
\ 2 /
$$\frac{\left(\sqrt{\frac{1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}} - \sqrt{\frac{1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}}\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
/ ____________ ____________\
| / 1 + sin(a) / 1 - sin(a) | /a\ /a pi\
| / ---------- - / ---------- |*tan|-|*tan|- + --|
\\/ 1 - sin(a) \/ 1 + sin(a) / \2/ \2 4 /
--------------------------------------------------------
/ 2/a pi\\ 2/a\
|1 + tan |- + --||*sin |-|
\ \2 4 // \2/
$$\frac{\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}$$
/ ____________ ____________\
2/a\ / 2/a pi\\ / 2/a\\ | / 1 + sin(a) / 1 - sin(a) |
sin |-|*|1 + tan |- + --||*|-1 + cot |-||*| / ---------- - / ---------- |
\2/ \ \2 4 // \ \2// \\/ 1 - sin(a) \/ 1 + sin(a) /
-------------------------------------------------------------------------------
2/a pi\
-1 + tan |- + --|
\2 4 /
$$\frac{\left(\sqrt{\frac{\sin{\left(a \right)} + 1}{- \sin{\left(a \right)} + 1}} - \sqrt{\frac{- \sin{\left(a \right)} + 1}{\sin{\left(a \right)} + 1}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}$$
/ _________________________ _________________________\ | / /a\ / /a\ | | / 1 + (1 + cos(a))*tan|-| / 1 - (1 + cos(a))*tan|-| | | / \2/ / \2/ | | / ----------------------- - / ----------------------- |*cos(a)*csc(a) | / /a\ / /a\ | | / 1 - (1 + cos(a))*tan|-| / 1 + (1 + cos(a))*tan|-| | \\/ \2/ \/ \2/ /
$$\left(\sqrt{\frac{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}{- \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}} - \sqrt{\frac{- \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}}\right) \cos{\left(a \right)} \csc{\left(a \right)}$$
/ _________________________ _________________________\
| / /a\ / /a\ |
| / 1 + (1 + cos(a))*tan|-| / 1 - (1 + cos(a))*tan|-| |
| / \2/ / \2/ |
| / ----------------------- - / ----------------------- |*cos(a)
| / /a\ / /a\ |
| / 1 - (1 + cos(a))*tan|-| / 1 + (1 + cos(a))*tan|-| |
\\/ \2/ \/ \2/ /
------------------------------------------------------------------------------
sin(a)
$$\frac{\left(\sqrt{\frac{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}{- \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}} - \sqrt{\frac{- \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 1}}\right) \cos{\left(a \right)}}{\sin{\left(a \right)}}$$
_________________ _________________
/ /a\ / /a\
/ 2*tan|-| / 2*tan|-|
/ \2/ / \2/
/ 1 + ----------- / 1 - -----------
/ 2/a\ / 2/a\
/ 1 + tan |-| / 1 + tan |-|
/ \2/ / \2/
/ --------------- - / ---------------
/ /a\ / /a\
/ 2*tan|-| / 2*tan|-|
/ \2/ / \2/
/ 1 - ----------- / 1 + -----------
/ 2/a\ / 2/a\
/ 1 + tan |-| / 1 + tan |-|
\/ \2/ \/ \2/
---------------------------------------------------------------------
tan(a)
$$\frac{\sqrt{\frac{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}} - \sqrt{\frac{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}}}{\tan{\left(a \right)}}$$
/ _________________ _________________\
| / /a\ / /a\ |
| / 2*tan|-| / 2*tan|-| |
| / \2/ / \2/ |
| / 1 + ----------- / 1 - ----------- |
| / 2/a\ / 2/a\ |
| / 1 + tan |-| / 1 + tan |-| |
/ 2/a\\ | / \2/ / \2/ |
|1 - tan |-||*| / --------------- - / --------------- |
\ \2// | / /a\ / /a\ |
| / 2*tan|-| / 2*tan|-| |
| / \2/ / \2/ |
| / 1 - ----------- / 1 + ----------- |
| / 2/a\ / 2/a\ |
| / 1 + tan |-| / 1 + tan |-| |
\\/ \2/ \/ \2/ /
-------------------------------------------------------------------------------------
/a\
2*tan|-|
\2/
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\sqrt{\frac{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}} - \sqrt{\frac{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}}\right)}{2 \tan{\left(\frac{a}{2} \right)}}$$
/ __________________________ __________________________\ | / 2 / 2 | | / 1 + -------------------- / 1 - -------------------- | | / / 1 \ /a\ / / 1 \ /a\ | | / |1 + -------|*cot|-| / |1 + -------|*cot|-| | | / | 2/a\| \2/ / | 2/a\| \2/ | | / | cot |-|| / | cot |-|| | | / \ \2// / \ \2// | | / ------------------------ - / ------------------------ |*cot(a) | / 2 / 2 | | / 1 - -------------------- / 1 + -------------------- | | / / 1 \ /a\ / / 1 \ /a\ | | / |1 + -------|*cot|-| / |1 + -------|*cot|-| | | / | 2/a\| \2/ / | 2/a\| \2/ | | / | cot |-|| / | cot |-|| | \\/ \ \2// \/ \ \2// /
$$\left(\sqrt{\frac{1 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}{1 - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}} - \sqrt{\frac{1 - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}{1 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}}\right) \cot{\left(a \right)}$$
/ _________________ _________________\
| / /a\ / /a\ |
| / 2*tan|-| / 2*tan|-| |
| / \2/ / \2/ |
| / 1 + ----------- / 1 - ----------- |
| / 2/a\ / 2/a\ |
2 | / 1 + tan |-| / 1 + tan |-| |
/ 2/a\\ | / \2/ / \2/ |
|1 + tan |-|| *| / --------------- - / --------------- |*tan(a)
\ \2// | / /a\ / /a\ |
| / 2*tan|-| / 2*tan|-| |
| / \2/ / \2/ |
| / 1 - ----------- / 1 + ----------- |
| / 2/a\ / 2/a\ |
| / 1 + tan |-| / 1 + tan |-| |
\\/ \2/ \/ \2/ /
---------------------------------------------------------------------------------------------
/ 2 \ 2/a\
4*\1 + tan (a)/*tan |-|
\2/
$$\frac{\left(\sqrt{\frac{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}} - \sqrt{\frac{1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \tan{\left(a \right)}}{4 \left(\tan^{2}{\left(a \right)} + 1\right) \tan^{2}{\left(\frac{a}{2} \right)}}$$
_________________________________ _________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ 1 + |< | / 1 - |< |
/ \\sin(a) otherwise / / \\sin(a) otherwise /
/ ------------------------------- - / -------------------------------
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ 1 - |< | / 1 + |< |
\/ \\sin(a) otherwise / \/ \\sin(a) otherwise /
-------------------------------------------------------------------------------------
tan(a)
$$\frac{\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)}{\tan{\left(a \right)}}$$
/ _________________________________ _________________________________\ | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / 1 + |< | / 1 - |< | | | / \\sin(a) otherwise / / \\sin(a) otherwise / | | / ------------------------------- - / ------------------------------- |*cot(a) | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / 1 - |< | / 1 + |< | | \\/ \\sin(a) otherwise / \/ \\sin(a) otherwise / /
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cot{\left(a \right)}$$
/ _________________ _________________\
| / /a\ / /a\ |
| / 2*cot|-| / 2*cot|-| |
| / \2/ / \2/ |
| / 1 + ----------- / 1 - ----------- |
| / 2/a\ / 2/a\ |
| / 1 + cot |-| / 1 + cot |-| |
/ 2/a\\ | / \2/ / \2/ | /a pi\
|1 + cot |-||*| / --------------- - / --------------- |*tan|- + --|
\ \2// | / /a\ / /a\ | \2 4 /
| / 2*cot|-| / 2*cot|-| |
| / \2/ / \2/ |
| / 1 - ----------- / 1 + ----------- |
| / 2/a\ / 2/a\ |
| / 1 + cot |-| / 1 + cot |-| |
\\/ \2/ \/ \2/ /
-------------------------------------------------------------------------------------------------
/ 2/a pi\\ /a\
|1 + tan |- + --||*cot|-|
\ \2 4 // \2/
$$\frac{\left(\sqrt{\frac{1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}{1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}} - \sqrt{\frac{1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}{1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}$$
_________________________________ _________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ 1 - |< | / 1 + |< |
/ \\sin(a) otherwise / / \\sin(a) otherwise /
- 2* / ------------------------------- + 2* / -------------------------------
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ 1 + |< | / 1 - |< |
\/ \\sin(a) otherwise / \/ \\sin(a) otherwise /
-------------------------------------------------------------------------------------------
2*tan(a)
$$\frac{\left(2 \left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) - \left(2 \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right)}{2 \tan{\left(a \right)}}$$
/ _________________________________ _________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / 1 + |< | / 1 - |< | |
| / \\sin(a) otherwise / / \\sin(a) otherwise / |
| / ------------------------------- - / ------------------------------- |*sin(2*a)
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / 1 - |< | / 1 + |< | |
\\/ \\sin(a) otherwise / \/ \\sin(a) otherwise / /
------------------------------------------------------------------------------------------------
2
2*sin (a)
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
/ ________________________________________ \
| / / 2/a\ \ |
| / | 4*sin |-|*sin(a) | |
| / | \2/ | |
| / (1 - cos(a))*|1 + -------------------| |
| / | 2 4/a\| _____________________ |
| / | sin (a) + 4*sin |-|| _____________________________________ / 2 |
| ___ / \ \2// / -1 / sin (a) 4/a\ | ___ / pi\||
|\/ 2 * / -------------------------------------- - / ----------------------------------- * / ------- + 2*sin |-| *|-1 + \/ 2 *sin|a + --|||*sin(2*a)
| / 2 / 2 \/ 2 \2/ | \ 4 /||
| / / ___ / pi\\ / / 2/a\ \ |
| / |-1 + \/ 2 *sin|a + --|| / (-1 + cos(a))*|2*sin |-| + sin(a)| |
\ \/ \ \ 4 // \/ \ \2/ / /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
2*sin (a)
$$\frac{\left(- \sqrt{- \frac{1}{\left(2 \sin^{2}{\left(\frac{a}{2} \right)} + \sin{\left(a \right)}\right)^{2} \left(\cos{\left(a \right)} - 1\right)}} \sqrt{2 \sin^{4}{\left(\frac{a}{2} \right)} + \frac{\sin^{2}{\left(a \right)}}{2}} \left|{\sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} - 1}\right| + \sqrt{2} \sqrt{\frac{\left(\frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + 1\right) \left(- \cos{\left(a \right)} + 1\right)}{\left(\sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} - 1\right)^{2}}}\right) \sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
/ _________________________________ _________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / 1 + |< 1 | / 1 - |< 1 | |
| / ||------ otherwise | / ||------ otherwise | |
| / \\csc(a) / / \\csc(a) / |
| / ------------------------------- - / ------------------------------- |*csc(a)
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / 1 - |< 1 | / 1 + |< 1 | |
| / ||------ otherwise | / ||------ otherwise | |
\\/ \\csc(a) / \/ \\csc(a) / /
------------------------------------------------------------------------------------------------------
/pi \
csc|-- - a|
\2 /
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ ______________________________________ ______________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / 1 + |< / pi\ | / 1 - |< / pi\ | |
| / ||cos|a - --| otherwise | / ||cos|a - --| otherwise | |
| / \\ \ 2 / / / \\ \ 2 / / |
| / ------------------------------------ - / ------------------------------------ |*cos(a)
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / 1 - |< / pi\ | / 1 + |< / pi\ | |
| / ||cos|a - --| otherwise | / ||cos|a - --| otherwise | |
\\/ \\ \ 2 / / \/ \\ \ 2 / / /
----------------------------------------------------------------------------------------------------------------
/ pi\
cos|a - --|
\ 2 /
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ ______________________________________ ______________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || 1 | / || 1 | |
| / 1 + |<----------- otherwise | / 1 - |<----------- otherwise | |
| / || / pi\ | / || / pi\ | |
| / ||sec|a - --| | / ||sec|a - --| | |
| / \\ \ 2 / / / \\ \ 2 / / | / pi\
| / ------------------------------------ - / ------------------------------------ |*sec|a - --|
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | \ 2 /
| / || | / || | |
| / || 1 | / || 1 | |
| / 1 - |<----------- otherwise | / 1 + |<----------- otherwise | |
| / || / pi\ | / || / pi\ | |
| / ||sec|a - --| | / ||sec|a - --| | |
\\/ \\ \ 2 / / \/ \\ \ 2 / / /
-----------------------------------------------------------------------------------------------------------------------------
sec(a)
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
/ ____________________________ ____________________________\
| / 2/a\ / 2/a\ |
| / 4*sin |-| / 4*sin |-| |
| / \2/ / \2/ |
| / 1 + ---------------------- / 1 - ---------------------- |
| / / 4/a\\ / / 4/a\\ |
| / | 4*sin |-|| / | 4*sin |-|| |
| / | \2/| / | \2/| |
| / |1 + ---------|*sin(a) / |1 + ---------|*sin(a) |
| / | 2 | / | 2 | |
| / \ sin (a) / / \ sin (a) / |
| / -------------------------- - / -------------------------- |*sin(2*a)
| / 2/a\ / 2/a\ |
| / 4*sin |-| / 4*sin |-| |
| / \2/ / \2/ |
| / 1 - ---------------------- / 1 + ---------------------- |
| / / 4/a\\ / / 4/a\\ |
| / | 4*sin |-|| / | 4*sin |-|| |
| / | \2/| / | \2/| |
| / |1 + ---------|*sin(a) / |1 + ---------|*sin(a) |
| / | 2 | / | 2 | |
\\/ \ sin (a) / \/ \ sin (a) / /
------------------------------------------------------------------------------------------------------------------
2
2*sin (a)
$$\frac{\left(\sqrt{\frac{1 + \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}{1 - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}} - \sqrt{\frac{1 - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}{1 + \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}}\right) \sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
_____________________________________ _____________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ ||1 - cos(a) | / ||1 - cos(a) |
/ 1 + |<---------- otherwise | / 1 - |<---------- otherwise |
/ || /a\ | / || /a\ |
/ || tan|-| | / || tan|-| |
/ \\ \2/ / / \\ \2/ /
/ ----------------------------------- - / -----------------------------------
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ ||1 - cos(a) | / ||1 - cos(a) |
/ 1 - |<---------- otherwise | / 1 + |<---------- otherwise |
/ || /a\ | / || /a\ |
/ || tan|-| | / || tan|-| |
\/ \\ \2/ / \/ \\ \2/ /
-------------------------------------------------------------------------------------------------------------
tan(a)
$$\frac{\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)}{\tan{\left(a \right)}}$$
/ _____________________________________ _____________________________________\
| / / 2/a pi\\ / / 2/a pi\\ |
| / |1 - cot |- + --||*(1 + sin(a)) / |1 - cot |- + --||*(1 + sin(a)) |
| / \ \2 4 // / \ \2 4 // |
| / 1 + ------------------------------- / 1 - ------------------------------- |
2/a\ / 2/a\\ | / 2 / 2 |
cos |-|*|1 - tan |-||*| / ----------------------------------- - / ----------------------------------- |
\2/ \ \2// | / / 2/a pi\\ / / 2/a pi\\ |
| / |1 - cot |- + --||*(1 + sin(a)) / |1 - cot |- + --||*(1 + sin(a)) |
| / \ \2 4 // / \ \2 4 // |
| / 1 - ------------------------------- / 1 + ------------------------------- |
\\/ 2 \/ 2 /
-----------------------------------------------------------------------------------------------------------------------------
/ 2/a pi\\ 2/a pi\
|1 - cot |- + --||*sin |- + --|
\ \2 4 // \2 4 /
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\sqrt{\frac{\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + 1}{- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + 1}} - \sqrt{\frac{- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + 1}{\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + 1}}\right) \cos^{2}{\left(\frac{a}{2} \right)}}{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ _______________________ _______________________\
| / 2/a pi\ / 2/a pi\ |
| / -1 + tan |- + --| / -1 + tan |- + --| |
| / \2 4 / / \2 4 / |
| / 1 + ----------------- / 1 - ----------------- |
| / 2/a pi\ / 2/a pi\ |
| / 1 + tan |- + --| / 1 + tan |- + --| |
/ 2/a pi\\ / 2/a\\ | / \2 4 / / \2 4 / |
|1 + tan |- + --||*|-1 + cot |-||*| / --------------------- - / --------------------- |
\ \2 4 // \ \2// | / 2/a pi\ / 2/a pi\ |
| / -1 + tan |- + --| / -1 + tan |- + --| |
| / \2 4 / / \2 4 / |
| / 1 - ----------------- / 1 + ----------------- |
| / 2/a pi\ / 2/a pi\ |
| / 1 + tan |- + --| / 1 + tan |- + --| |
\\/ \2 4 / \/ \2 4 / /
---------------------------------------------------------------------------------------------------------------------
/ 2/a\\ / 2/a pi\\
|1 + cot |-||*|-1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{\left(\sqrt{\frac{\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}{- \frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}} - \sqrt{\frac{- \frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}{\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
______________________________________ ______________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ || /a\ | / || /a\ |
/ || 2*tan|-| | / || 2*tan|-| |
/ 1 + |< \2/ | / 1 - |< \2/ |
/ ||----------- otherwise | / ||----------- otherwise |
/ || 2/a\ | / || 2/a\ |
/ ||1 + tan |-| | / ||1 + tan |-| |
/ \\ \2/ / / \\ \2/ /
/ ------------------------------------ - / ------------------------------------
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ || /a\ | / || /a\ |
/ || 2*tan|-| | / || 2*tan|-| |
/ 1 - |< \2/ | / 1 + |< \2/ |
/ ||----------- otherwise | / ||----------- otherwise |
/ || 2/a\ | / || 2/a\ |
/ ||1 + tan |-| | / ||1 + tan |-| |
\/ \\ \2/ / \/ \\ \2/ /
-----------------------------------------------------------------------------------------------------------------------
tan(a)
$$\frac{\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)}{\tan{\left(a \right)}}$$
/ ______________________ ______________________\
| / 2/a pi\ / 2/a pi\ |
| / 1 - cot |- + --| / 1 - cot |- + --| |
| / \2 4 / / \2 4 / |
| / 1 + ---------------- / 1 - ---------------- |
| / 2/a pi\ / 2/a pi\ |
| / 1 + cot |- + --| / 1 + cot |- + --| |
/ 2/a pi\\ / 2/a\\ | / \2 4 / / \2 4 / |
|1 + cot |- + --||*|1 - tan |-||*| / -------------------- - / -------------------- |
\ \2 4 // \ \2// | / 2/a pi\ / 2/a pi\ |
| / 1 - cot |- + --| / 1 - cot |- + --| |
| / \2 4 / / \2 4 / |
| / 1 - ---------------- / 1 + ---------------- |
| / 2/a pi\ / 2/a pi\ |
| / 1 + cot |- + --| / 1 + cot |- + --| |
\\/ \2 4 / \/ \2 4 / /
------------------------------------------------------------------------------------------------------------------
/ 2/a\\ / 2/a pi\\
|1 + tan |-||*|1 - cot |- + --||
\ \2// \ \2 4 //
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\sqrt{\frac{\frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}{- \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}} - \sqrt{\frac{- \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}{\frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ ______________________________________ ______________________________________\ | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / || | / || | | | / || /a\ | / || /a\ | | | / || 2*cot|-| | / || 2*cot|-| | | | / 1 + |< \2/ | / 1 - |< \2/ | | | / ||----------- otherwise | / ||----------- otherwise | | | / || 2/a\ | / || 2/a\ | | | / ||1 + cot |-| | / ||1 + cot |-| | | | / \\ \2/ / / \\ \2/ / | | / ------------------------------------ - / ------------------------------------ |*cot(a) | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / || | / || | | | / || /a\ | / || /a\ | | | / || 2*cot|-| | / || 2*cot|-| | | | / 1 - |< \2/ | / 1 + |< \2/ | | | / ||----------- otherwise | / ||----------- otherwise | | | / || 2/a\ | / || 2/a\ | | | / ||1 + cot |-| | / ||1 + cot |-| | | \\/ \\ \2/ / \/ \\ \2/ / /
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cot{\left(a \right)}$$
/ _________________________________ _________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | // zoo for a mod pi = 0\
| / 1 + |< | / 1 - |< | | || |
| / \\sin(a) otherwise / / \\sin(a) otherwise / | // 0 for 2*a mod pi = 0\ || 1 |
| / ------------------------------- - / ------------------------------- |*|< |*|<------- otherwise |
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | \\sin(2*a) otherwise / || 2 |
| / 1 - |< | / 1 + |< | | ||sin (a) |
\\/ \\sin(a) otherwise / \/ \\sin(a) otherwise / / \\ /
----------------------------------------------------------------------------------------------------------------------------------------------------
2
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)}{2}$$
/ ____________________________________________________ ____________________________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / 1 + |
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cot{\left(a \right)}$$
_______________________________________________ _______________________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ || 2 | / || 2 |
/ ||-------------------- otherwise | / ||-------------------- otherwise |
/ 1 + |
$$\frac{\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)}{\tan{\left(a \right)}}$$
/ ____________________________________ ____________________________________\
| / /a\ / /a\ |
| / 2*sec|-| / 2*sec|-| |
| / \2/ / \2/ |
| / 1 + ------------------------------ / 1 - ------------------------------ |
| / / 2/a\ \ / / 2/a\ \ |
| / | sec |-| | / | sec |-| | |
| / | \2/ | /a pi\ / | \2/ | /a pi\ |
| / |1 + ------------|*sec|- - --| / |1 + ------------|*sec|- - --| |
| / | 2/a pi\| \2 2 / / | 2/a pi\| \2 2 / |
| / | sec |- - --|| / | sec |- - --|| |
| / \ \2 2 // / \ \2 2 // | / pi\
| / ---------------------------------- - / ---------------------------------- |*sec|a - --|
| / /a\ / /a\ | \ 2 /
| / 2*sec|-| / 2*sec|-| |
| / \2/ / \2/ |
| / 1 - ------------------------------ / 1 + ------------------------------ |
| / / 2/a\ \ / / 2/a\ \ |
| / | sec |-| | / | sec |-| | |
| / | \2/ | /a pi\ / | \2/ | /a pi\ |
| / |1 + ------------|*sec|- - --| / |1 + ------------|*sec|- - --| |
| / | 2/a pi\| \2 2 / / | 2/a pi\| \2 2 / |
| / | sec |- - --|| / | sec |- - --|| |
\\/ \ \2 2 // \/ \ \2 2 // /
-----------------------------------------------------------------------------------------------------------------------------------------
sec(a)
$$\frac{\left(\sqrt{\frac{1 + \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{1 - \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}} - \sqrt{\frac{1 - \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{1 + \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
/ _______________________________ _______________________________\
| / /a pi\ / /a pi\ |
| / 2*cos|- - --| / 2*cos|- - --| |
| / \2 2 / / \2 2 / |
| / 1 + ------------------------- / 1 - ------------------------- |
| / / 2/a pi\\ / / 2/a pi\\ |
| / | cos |- - --|| / | cos |- - --|| |
| / | \2 2 /| /a\ / | \2 2 /| /a\ |
| / |1 + ------------|*cos|-| / |1 + ------------|*cos|-| |
| / | 2/a\ | \2/ / | 2/a\ | \2/ |
| / | cos |-| | / | cos |-| | |
| / \ \2/ / / \ \2/ / |
| / ----------------------------- - / ----------------------------- |*cos(a)
| / /a pi\ / /a pi\ |
| / 2*cos|- - --| / 2*cos|- - --| |
| / \2 2 / / \2 2 / |
| / 1 - ------------------------- / 1 + ------------------------- |
| / / 2/a pi\\ / / 2/a pi\\ |
| / | cos |- - --|| / | cos |- - --|| |
| / | \2 2 /| /a\ / | \2 2 /| /a\ |
| / |1 + ------------|*cos|-| / |1 + ------------|*cos|-| |
| / | 2/a\ | \2/ / | 2/a\ | \2/ |
| / | cos |-| | / | cos |-| | |
\\/ \ \2/ / \/ \ \2/ / /
--------------------------------------------------------------------------------------------------------------------------
/ pi\
cos|a - --|
\ 2 /
$$\frac{\left(\sqrt{\frac{1 + \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}{1 - \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}} - \sqrt{\frac{1 - \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}{1 + \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}}\right) \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ _______________________________ _______________________________\
| / /pi a\ / /pi a\ |
| / 2*csc|-- - -| / 2*csc|-- - -| |
| / \2 2/ / \2 2/ |
| / 1 + ------------------------- / 1 - ------------------------- |
| / / 2/pi a\\ / / 2/pi a\\ |
| / | csc |-- - -|| / | csc |-- - -|| |
| / | \2 2/| /a\ / | \2 2/| /a\ |
| / |1 + ------------|*csc|-| / |1 + ------------|*csc|-| |
| / | 2/a\ | \2/ / | 2/a\ | \2/ |
| / | csc |-| | / | csc |-| | |
| / \ \2/ / / \ \2/ / |
| / ----------------------------- - / ----------------------------- |*csc(a)
| / /pi a\ / /pi a\ |
| / 2*csc|-- - -| / 2*csc|-- - -| |
| / \2 2/ / \2 2/ |
| / 1 - ------------------------- / 1 + ------------------------- |
| / / 2/pi a\\ / / 2/pi a\\ |
| / | csc |-- - -|| / | csc |-- - -|| |
| / | \2 2/| /a\ / | \2 2/| /a\ |
| / |1 + ------------|*csc|-| / |1 + ------------|*csc|-| |
| / | 2/a\ | \2/ / | 2/a\ | \2/ |
| / | csc |-| | / | csc |-| | |
\\/ \ \2/ / \/ \ \2/ / /
--------------------------------------------------------------------------------------------------------------------------
/pi \
csc|-- - a|
\2 /
$$\frac{\left(\sqrt{\frac{1 + \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}{1 - \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}} - \sqrt{\frac{1 - \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}{1 + \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}}\right) \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ ____________________________________________ ____________________________________________\ | / // / 3*pi\ \ / // / 3*pi\ \ | // / 3*pi\ \ | / || 1 for |a + ----| mod 2*pi = 0| / || 1 for |a + ----| mod 2*pi = 0| | || 1 for |a + ----| mod 2*pi = 0| | / 1 + |< \ 2 / | / 1 - |< \ 2 / | | || \ 2 / | | / || | / || | | || | | / \\sin(a) otherwise / / \\sin(a) otherwise / | // 1 for a mod 2*pi = 0\ || 1 /a\ | | / ------------------------------------------ - / ------------------------------------------ |*|< |*|<------ + tan|-| | | / // / 3*pi\ \ / // / 3*pi\ \ | \\cos(a) otherwise / || /a\ \2/ | | / || 1 for |a + ----| mod 2*pi = 0| / || 1 for |a + ----| mod 2*pi = 0| | ||tan|-| | | / 1 - |< \ 2 / | / 1 + |< \ 2 / | | || \2/ | | / || | / || | | ||--------------- otherwise | \\/ \\sin(a) otherwise / \/ \\sin(a) otherwise / / \\ 2 /
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}}{2} & \text{otherwise} \end{cases}\right)$$
/ _________________________________________________________ _________________________________________________________\ | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / || | / || | | | / ||/ 0 for a mod pi = 0 | / ||/ 0 for a mod pi = 0 | | | / ||| | / ||| | | | / ||| /a\ | / ||| /a\ | | | / 1 + |<| 2*cot|-| | / 1 - |<| 2*cot|-| | | | / ||< \2/ otherwise | / ||< \2/ otherwise | | | / |||----------- otherwise | / |||----------- otherwise | | | / ||| 2/a\ | / ||| 2/a\ | | | / |||1 + cot |-| | / |||1 + cot |-| | | | / \\\ \2/ / / \\\ \2/ / | | / ------------------------------------------------------- - / ------------------------------------------------------- |*cot(a) | / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | | / || | / || | | | / ||/ 0 for a mod pi = 0 | / ||/ 0 for a mod pi = 0 | | | / ||| | / ||| | | | / ||| /a\ | / ||| /a\ | | | / 1 - |<| 2*cot|-| | / 1 + |<| 2*cot|-| | | | / ||< \2/ otherwise | / ||< \2/ otherwise | | | / |||----------- otherwise | / |||----------- otherwise | | | / ||| 2/a\ | / ||| 2/a\ | | | / |||1 + cot |-| | / |||1 + cot |-| | | \\/ \\\ \2/ / \/ \\\ \2/ / /
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cot{\left(a \right)}$$
_________________________________________________________ _________________________________________________________
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ || 2*(-sin(2*a) + 2*sin(a)) | / || 2*(-sin(2*a) + 2*sin(a)) |
/ 1 - |<------------------------------ otherwise | / 1 + |<------------------------------ otherwise |
/ || 2 | / || 2 |
/ ||1 - cos(2*a) + 2*(1 - cos(a)) | / ||1 - cos(2*a) + 2*(1 - cos(a)) |
/ \\ / / \\ /
- 2* / ------------------------------------------------------- + 2* / -------------------------------------------------------
/ // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\
/ || | / || |
/ || 2*(-sin(2*a) + 2*sin(a)) | / || 2*(-sin(2*a) + 2*sin(a)) |
/ 1 + |<------------------------------ otherwise | / 1 - |<------------------------------ otherwise |
/ || 2 | / || 2 |
/ ||1 - cos(2*a) + 2*(1 - cos(a)) | / ||1 - cos(2*a) + 2*(1 - cos(a)) |
\/ \\ / \/ \\ /
-----------------------------------------------------------------------------------------------------------------------------------------------------------
2*tan(a)
$$\frac{\left(2 \left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(a \right)} - \sin{\left(2 a \right)}\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(a \right)} - \sin{\left(2 a \right)}\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) - \left(2 \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(a \right)} - \sin{\left(2 a \right)}\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(a \right)} - \sin{\left(2 a \right)}\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)\right)}{2 \tan{\left(a \right)}}$$
/ __________________________________________________ __________________________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || sin(a) | / || sin(a) | |
| / ||----------------------- otherwise | / ||----------------------- otherwise | |
| / ||/ 2 \ | / ||/ 2 \ | |
| / 1 + |<| sin (a) | 2/a\ | / 1 - |<| sin (a) | 2/a\ | |
| / |||1 + ---------|*sin |-| | / |||1 + ---------|*sin |-| | |
| / ||| 4/a\| \2/ | / ||| 4/a\| \2/ | |
| / ||| 4*sin |-|| | / ||| 4*sin |-|| | |
| / ||\ \2// | / ||\ \2// | |
| / \\ / / \\ / |
| / ------------------------------------------------ - / ------------------------------------------------ |*sin(2*a)
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || sin(a) | / || sin(a) | |
| / ||----------------------- otherwise | / ||----------------------- otherwise | |
| / ||/ 2 \ | / ||/ 2 \ | |
| / 1 - |<| sin (a) | 2/a\ | / 1 + |<| sin (a) | 2/a\ | |
| / |||1 + ---------|*sin |-| | / |||1 + ---------|*sin |-| | |
| / ||| 4/a\| \2/ | / ||| 4/a\| \2/ | |
| / ||| 4*sin |-|| | / ||| 4*sin |-|| | |
| / ||\ \2// | / ||\ \2// | |
\\/ \\ / \/ \\ / /
------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
2*sin (a)
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
/ ______________________________________ ______________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || /a\ | / || /a\ | |
| / || 2*cot|-| | / || 2*cot|-| | |
| / 1 + |< \2/ | / 1 - |< \2/ | | // zoo for a mod pi = 0\
| / ||----------- otherwise | / ||----------- otherwise | | || |
| / || 2/a\ | / || 2/a\ | | // 0 for 2*a mod pi = 0\ || 2 |
| / ||1 + cot |-| | / ||1 + cot |-| | | || | ||/ 2/a\\ |
| / \\ \2/ / / \\ \2/ / | || 2*cot(a) | |||1 + cot |-|| |
| / ------------------------------------ - / ------------------------------------ |*|<----------- otherwise |*|<\ \2// |
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ | || 2 | ||-------------- otherwise |
| / || | / || | | ||1 + cot (a) | || 2/a\ |
| / || /a\ | / || /a\ | | \\ / || 4*cot |-| |
| / || 2*cot|-| | / || 2*cot|-| | | || \2/ |
| / 1 - |< \2/ | / 1 + |< \2/ | | \\ /
| / ||----------- otherwise | / ||----------- otherwise | |
| / || 2/a\ | / || 2/a\ | |
| / ||1 + cot |-| | / ||1 + cot |-| | |
\\/ \\ \2/ / \/ \\ \2/ / /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{2}$$
/ _________________________________________________________ _________________________________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || /a\ | / || /a\ | |
| / || 2*cos|-| | / || 2*cos|-| | |
| / || \2/ | / || \2/ | |
| / ||------------------------------ otherwise | / ||------------------------------ otherwise | |
| / 1 + |
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ ____________________________________________________ ____________________________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || /a pi\ | / || /a pi\ | |
| / || 2*sec|- - --| | / || 2*sec|- - --| | |
| / || \2 2 / | / || \2 2 / | |
| / ||------------------------- otherwise | / ||------------------------- otherwise | |
| / 1 + |
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
/ _________________________________________________________ _________________________________________________________\
| / // 0 for a mod pi = 0\ / // 0 for a mod pi = 0\ |
| / || | / || | |
| / || /a\ | / || /a\ | |
| / || 2*csc|-| | / || 2*csc|-| | |
| / || \2/ | / || \2/ | |
| / ||------------------------------ otherwise | / ||------------------------------ otherwise | |
| / 1 + |
$$\frac{\left(\left(\sqrt{\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ _______________________________________________________ _______________________________________________________\ | / // / 3*pi\ \ / // / 3*pi\ \ | | / || 1 for |a + ----| mod 2*pi = 0| / || 1 for |a + ----| mod 2*pi = 0| | | / || \ 2 / | / || \ 2 / | | | / || | / || | | | / || 2/a pi\ | / || 2/a pi\ | | | / 1 + |<-1 + tan |- + --| | / 1 - |<-1 + tan |- + --| | | | / || \2 4 / | / || \2 4 / | | // / 3*pi\ \ | / ||----------------- otherwise | / ||----------------- otherwise | | // 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0| | / || 2/a pi\ | / || 2/a pi\ | | || | || \ 2 / | | / || 1 + tan |- + --| | / || 1 + tan |- + --| | | || 2/a\ | || | | / \\ \2 4 / / / \\ \2 4 / / | ||-1 + cot |-| | || 2/a pi\ | | / ----------------------------------------------------- - / ----------------------------------------------------- |*|< \2/ |*|< 1 + tan |- + --| | | / // / 3*pi\ \ / // / 3*pi\ \ | ||------------ otherwise | || \2 4 / | | / || 1 for |a + ----| mod 2*pi = 0| / || 1 for |a + ----| mod 2*pi = 0| | || 2/a\ | ||----------------- otherwise | | / || \ 2 / | / || \ 2 / | | ||1 + cot |-| | || 2/a pi\ | | / || | / || | | \\ \2/ / ||-1 + tan |- + --| | | / || 2/a pi\ | / || 2/a pi\ | | \\ \2 4 / / | / 1 - |<-1 + tan |- + --| | / 1 + |<-1 + tan |- + --| | | | / || \2 4 / | / || \2 4 / | | | / ||----------------- otherwise | / ||----------------- otherwise | | | / || 2/a pi\ | / || 2/a pi\ | | | / || 1 + tan |- + --| | / || 1 + tan |- + --| | | \\/ \\ \2 4 / / \/ \\ \2 4 / / /
$$\left(\left(\sqrt{\frac{\left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right) - \left(\sqrt{\frac{\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}{\left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)$$
(sqrt((1 + Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + tan(a/2 + pi/4)^2)/(1 + tan(a/2 + pi/4)^2), True)))/(1 - Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + tan(a/2 + pi/4)^2)/(1 + tan(a/2 + pi/4)^2), True)))) - sqrt((1 - Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + tan(a/2 + pi/4)^2)/(1 + tan(a/2 + pi/4)^2), True)))/(1 + Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + tan(a/2 + pi/4)^2)/(1 + tan(a/2 + pi/4)^2), True)))))*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + cot(a/2)^2)/(1 + cot(a/2)^2), True))*Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((1 + tan(a/2 + pi/4)^2)/(-1 + tan(a/2 + pi/4)^2), True))