Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sqrt((a-8)^2)+sqrt((a-9)^2) если a=1/2
- 1-2*cos(4*a)+2*cos(8*a)-2*cos(12*a) если a=1/2
- (3*m^6*n^3)^4*m^8*n^2/81 если m=-1/3
- cos(2*a)^4-6*cos(2*a)^2*sin(2*a)^2+sin(2*a)^4 если a=-2
- Разложить многочлен на множители:
- 81*a^6-64*b^4
- x^6-4096
- z^2+4*z+13
- 7*p^2-7
- Общий знаменатель:
- ((3*b^2+3)/(1-b))+((6*b)/(b-1))
- (a-5/(a^2-5*a+25)-(12*a-61)/(a^3+125))/(3*a-18/2*a^2-10*a+50)
- ((1/(t^2+3*t+2))+(2*t/(t^2+4*t+3))+(1/(t^2+5*t+6)))^2*((t-3)^2+12*t)/2
- (x-2/9+1/81*x)/(x-1/81*x)
- Умножение столбиком:
- 123*305
- 24*49
- 49*54
- 8902*80
- Деление столбиком:
- 242/7
- 93080/320
- 341/2
- 521/2
- Раскрыть скобки в:
- a*(a+2)*(a-2)-(a-1)*(a2+a+1)
- (a-3*b)*(a+3*b)-(a-3*b)*(a-3*b)
- 9*(a-1)*(a+2)
- 4*(t+1)^4+4*t^2+(t-1)^4
- Разложение числа на простые множители:
- 41595
- 5766
- 82691
- 2058
- График функции y =:
- -3
- Производная:
- -3
- Интеграл d{x}:
-
-3
-
Идентичные выражения
- девятнадцать +sin(четыре *a)-cos(четыре *a)+cos(два *a) если a=- три
- 19 плюс синус от (4 умножить на a) минус косинус от (4 умножить на a) плюс косинус от (2 умножить на a) если a равно минус 3
- девятнадцать плюс синус от (четыре умножить на a) минус косинус от (четыре умножить на a) плюс косинус от (два умножить на a) если a равно минус три
- 19+sin(4a)-cos(4a)+cos(2a) если a=-3
- 19+sin4a-cos4a+cos2a если a=-3
- 19+sin(4*a)-cos(4*a)+cos(2*a) при a=-3
-
Похожие выражения
- 19-sin(4*a)-cos(4*a)+cos(2*a) если a=-3
- 19+sin(4*a)-cos(4*a)+cos(2*a) если a=+3
- 19+sin(4*a)+cos(4*a)+cos(2*a) если a=-3
- 19+sin(4*a)-cos(4*a)-cos(2*a) если a=-3
19+sin(4*a)-cos(4*a)+cos(2*a) если a=-3
Решение
Вы ввели
[src]
19 + sin(4*a) - cos(4*a) + cos(2*a)
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} - \cos{\left(4 a \right)} + 19$$
19 + sin(4*a) - cos(4*a) + cos(2*a)
Подстановка условия
[src]
19 + sin(4*a) - cos(4*a) + cos(2*a) при a = -3
подставляем
19 + sin(4*a) - cos(4*a) + cos(2*a)
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} - \cos{\left(4 a \right)} + 19$$
19 - cos(4*a) + cos(2*a) + sin(4*a)
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} - \cos{\left(4 a \right)} + 19$$
переменные
a = -3
$$a = -3$$
19 - cos(4*(-3)) + cos(2*(-3)) + sin(4*(-3))
$$\sin{\left(4 (-3) \right)} + \cos{\left(2 (-3) \right)} - \cos{\left(4 (-3) \right)} + 19$$
19 - cos(12) - sin(12) + cos(6)
$$- \cos{\left(12 \right)} - \sin{\left(12 \right)} + \cos{\left(6 \right)} + 19$$
19 - cos(12) - sin(12) + cos(6)
Степени
[src]
-2*I*a 2*I*a -4*I*a 4*I*a / -4*I*a 4*I*a\
e e e e I*\- e + e /
19 + ------- + ------ - ------- - ------ - ----------------------
2 2 2 2 2
$$- \frac{e^{4 i a}}{2} + \frac{e^{2 i a}}{2} - \frac{i \left(e^{4 i a} - e^{- 4 i a}\right)}{2} + 19 + \frac{e^{- 2 i a}}{2} - \frac{e^{- 4 i a}}{2}$$
19 + exp(-2*i*a)/2 + exp(2*i*a)/2 - exp(-4*i*a)/2 - exp(4*i*a)/2 - i*(-exp(-4*i*a) + exp(4*i*a))/2
Тригонометрическая часть
[src]
19 + 2*sin(a)*sin(3*a) + sin(4*a)
$$2 \sin{\left(a \right)} \sin{\left(3 a \right)} + \sin{\left(4 a \right)} + 19$$
1
19 - -------- + cos(2*a) + sin(4*a)
sec(4*a)
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19 - \frac{1}{\sec{\left(4 a \right)}}$$
/ pi\
19 - cos(4*a) + cos(2*a) + cos|4*a - --|
\ 2 /
$$\cos{\left(2 a \right)} - \cos{\left(4 a \right)} + \cos{\left(4 a - \frac{\pi}{2} \right)} + 19$$
/pi \
19 - cos(4*a) + sin(4*a) + sin|-- + 2*a|
\2 /
$$\sin{\left(4 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} - \cos{\left(4 a \right)} + 19$$
/pi \
19 - sin|-- + 4*a| + cos(2*a) + sin(4*a)
\2 /
$$\sin{\left(4 a \right)} - \sin{\left(4 a + \frac{\pi}{2} \right)} + \cos{\left(2 a \right)} + 19$$
1
19 - ------------- + cos(2*a) + sin(4*a)
/pi \
csc|-- - 4*a|
\2 /
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19 - \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$
1 1 1
19 + -------- + -------- - --------
csc(4*a) sec(2*a) sec(4*a)
$$19 - \frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(4 a \right)}}$$
/pi \ /pi \
19 - sin|-- + 4*a| + sin(4*a) + sin|-- + 2*a|
\2 / \2 /
$$\sin{\left(4 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} - \sin{\left(4 a + \frac{\pi}{2} \right)} + 19$$
1 1 1
19 + -------- + ------------- - --------
sec(2*a) / pi\ sec(4*a)
sec|4*a - --|
\ 2 /
$$19 + \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
2
1 - tan (2*a)
19 - ------------- + cos(2*a) + sin(4*a)
2
sec (2*a)
$$\sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19 - \frac{- \tan^{2}{\left(2 a \right)} + 1}{\sec^{2}{\left(2 a \right)}}$$
2
-1 + 2*cos (2*a) + cos(4*a)
19 - --------------------------- + cos(2*a) + sin(4*a)
2
$$- \frac{2 \cos^{2}{\left(2 a \right)} + \cos{\left(4 a \right)} - 1}{2} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19$$
1 1 1
19 + -------- + ------------- - --------
sec(2*a) /pi \ sec(4*a)
sec|-- - 4*a|
\2 /
$$19 + \frac{1}{\sec{\left(- 4 a + \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
1 1 1
19 + -------- + ------------- - --------
csc(4*a) /pi \ sec(4*a)
csc|-- - 2*a|
\2 /
$$19 - \frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(4 a \right)}}$$
2 / 2 \ 19 - sin (2*a)*\-1 + cot (2*a)/ + cos(2*a) + sin(4*a)
$$- \left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19$$
2 / 2 \ 19 - cos (2*a)*\1 - tan (2*a)/ + cos(2*a) + sin(4*a)
$$- \left(- \tan^{2}{\left(2 a \right)} + 1\right) \cos^{2}{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19$$
1 1 1
19 + -------- + ------------- - -------------
csc(4*a) /pi \ /pi \
csc|-- - 2*a| csc|-- - 4*a|
\2 / \2 /
$$19 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(4 a \right)}}$$
1 1 1
19 + ------------- + ------------- - -------------
csc(pi - 4*a) /pi \ /pi \
csc|-- - 2*a| csc|-- - 4*a|
\2 / \2 /
$$19 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 4 a + \pi \right)}}$$
2 ___ 3 / pi\
19 + 6*sin (a) + 4*cos(a)*sin(a) - 8*\/ 2 *sin (a)*sin|a + --|
\ 4 /
$$- 8 \sqrt{2} \sin^{3}{\left(a \right)} \sin{\left(a + \frac{\pi}{4} \right)} + 6 \sin^{2}{\left(a \right)} + 4 \sin{\left(a \right)} \cos{\left(a \right)} + 19$$
2
/ 2 \ 4 / 2 \
19 - \1 - tan (a)/ *cos (a)*\1 - tan (2*a)/ + cos(2*a) + sin(4*a)
$$- \left(- \tan^{2}{\left(a \right)} + 1\right)^{2} \cdot \left(- \tan^{2}{\left(2 a \right)} + 1\right) \cos^{4}{\left(a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)} + 19$$
/ pi\
2 2*tan|a + --|
-1 + 2*cos (2*a) + cos(4*a) \ 4 /
19 - --------------------------- + ---------------- + sin(4*a)
2 2/ pi\
1 + tan |a + --|
\ 4 /
$$- \frac{2 \cos^{2}{\left(2 a \right)} + \cos{\left(4 a \right)} - 1}{2} + \sin{\left(4 a \right)} + 19 + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
/ 2/ pi\\
| cos |2*a - --||
2 | \ 2 /| / pi\
19 - cos (2*a)*|1 - --------------| + cos(2*a) + cos|4*a - --|
| 2 | \ 2 /
\ cos (2*a) /
$$- \left(1 - \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \cos^{2}{\left(2 a \right)} + \cos{\left(2 a \right)} + \cos{\left(4 a - \frac{\pi}{2} \right)} + 19$$
/ 4 \
2/pi \ | 4*sin (2*a)| /pi \
19 - sin |-- + 2*a|*|1 - -----------| + sin(4*a) + sin|-- + 2*a|
\2 / | 2 | \2 /
\ sin (4*a) /
$$- \left(- \frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \sin^{2}{\left(2 a + \frac{\pi}{2} \right)} + \sin{\left(4 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} + 19$$
2
sec (2*a)
1 - --------------
2/ pi\
sec |2*a - --|
1 1 \ 2 /
19 + -------- + ------------- - ------------------
sec(2*a) / pi\ 2
sec|4*a - --| sec (2*a)
\ 2 /
$$19 - \frac{- \frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1}{\sec^{2}{\left(2 a \right)}} + \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
2 2
1 - tan (a) 1 - tan (2*a) 2*tan(2*a)
19 + ----------- - ------------- + -------------
2 2 2
1 + tan (a) 1 + tan (2*a) 1 + tan (2*a)
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} - \frac{- \tan^{2}{\left(2 a \right)} + 1}{\tan^{2}{\left(2 a \right)} + 1} + 19 + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
2/pi \
csc |-- - 2*a|
\2 /
1 - --------------
2
1 1 csc (2*a)
19 + -------- + ------------- - ------------------
csc(4*a) /pi \ 2/pi \
csc|-- - 2*a| csc |-- - 2*a|
\2 / \2 /
$$19 - \frac{1 - \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(4 a \right)}}$$
/ pi\
2 2*tan|2*a + --|
1 - tan (a) \ 4 / 2*tan(2*a)
19 + ----------- - ------------------ + -------------
2 2/ pi\ 2
1 + tan (a) 1 + tan |2*a + --| 1 + tan (2*a)
\ 4 /
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + 19 - \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
2/ pi\
2 -1 + tan |2*a + --| 2
-1 + cot (a) \ 4 / -1 + cot (2*a)
19 + ------------ + ------------------- - --------------
2 2/ pi\ 2
1 + cot (a) 1 + tan |2*a + --| 1 + cot (2*a)
\ 4 /
$$\frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} - \frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} + 19$$
2/ pi\
1 - cot |2*a + --| 2 2
\ 4 / 1 - tan (a) 1 - tan (2*a)
19 + ------------------ + ----------- - -------------
2/ pi\ 2 2
1 + cot |2*a + --| 1 + tan (a) 1 + tan (2*a)
\ 4 /
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} - \frac{- \tan^{2}{\left(2 a \right)} + 1}{\tan^{2}{\left(2 a \right)} + 1} + \frac{- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + 19$$
/ pi\ / pi\
2*tan|2*a + --| 2*tan|a + --|
\ 4 / 2*tan(2*a) \ 4 /
19 - ------------------ + ------------- + ----------------
2/ pi\ 2 2/ pi\
1 + tan |2*a + --| 1 + tan (2*a) 1 + tan |a + --|
\ 4 / \ 4 /
$$19 - \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
/ pi\ / pi\
2*tan|2*a + --| 2*tan|a + --|
\ 4 / 2*cot(2*a) \ 4 /
19 - ------------------ + ------------- + ----------------
2/ pi\ 2 2/ pi\
1 + tan |2*a + --| 1 + cot (2*a) 1 + tan |a + --|
\ 4 / \ 4 /
$$19 + \frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} - \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - cos(4*a) + |< | + |< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \cos{\left(4 a \right)} + 19$$
1 1
1 - ------- 1 - ---------
2 2
cot (a) cot (2*a) 2
19 + ----------- - ------------- + ------------------------
1 1 / 1 \
1 + ------- 1 + --------- |1 + ---------|*cot(2*a)
2 2 | 2 |
cot (a) cot (2*a) \ cot (2*a)/
$$\frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} - \frac{1 - \frac{1}{\cot^{2}{\left(2 a \right)}}}{1 + \frac{1}{\cot^{2}{\left(2 a \right)}}} + 19 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \cot{\left(2 a \right)}}$$
2
2 / 2 \ / 2 \
1 - tan (a) 2*tan(2*a) \1 - tan (a)/ *\1 - tan (2*a)/
19 + ----------- + ------------- - ------------------------------
2 2 2
1 + tan (a) 1 + tan (2*a) / 2 \
\1 + tan (a)/
$$- \frac{\left(- \tan^{2}{\left(a \right)} + 1\right)^{2} \cdot \left(- \tan^{2}{\left(2 a \right)} + 1\right)}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} + \frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + 19 + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
2
2 / 2 2 \ 2 2 2 / 2 2 \
19 + cos (a) - \cos (a) - sin (a)/ - sin (a) + 4*cos (a)*sin (a) + 4*\cos (a) - sin (a)/*cos(a)*sin(a)
$$4 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + 4 \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right) \sin{\left(a \right)} \cos{\left(a \right)} - \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2} - \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + 19$$
// /pi \ \
|| 0 for |-- + 4*a| mod pi = 0| // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |< \2 / | + |< | + |< |
|| | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
\\cos(4*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
2 || | || |
-1 + cot (2*a) || 2*cot(2*a) | || 2 |
19 - -------------- + |<------------- otherwise | + |<-1 + cot (a) |
2 || 2 | ||------------ otherwise |
1 + cot (2*a) ||1 + cot (2*a) | || 2 |
\\ / \\1 + cot (a) /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} + 19$$
// 1 for 2*a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |< 2 / 2 \ | + |< | + |< |
||sin (2*a)*\-1 + cot (2*a)/ otherwise | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
\\ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 1 |
||-1 + --------- | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |< 2 | + |< | + |< |
|| tan (2*a) | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
||-------------- otherwise |
|| 2 |
\\ csc (2*a) /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 19$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 4*a| mod pi = 0| // 0 for 4*a mod pi = 0\ || 0 for |-- + 2*a| mod pi = 0|
19 - |< \2 / | + |< | + |< \2 / |
|| | \\sin(4*a) otherwise / || |
\\cos(4*a) otherwise / \\cos(2*a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 2 / 1 \ | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
/ pi\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
2*tan|2*a + --| || | || |
\ 4 / || 2*cot(2*a) | || 2 |
19 - ------------------ + |<------------- otherwise | + |<-1 + cot (a) |
2/ pi\ || 2 | ||------------ otherwise |
1 + tan |2*a + --| ||1 + cot (2*a) | || 2 |
\ 4 / \\ / \\1 + cot (a) /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 19 - \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}$$
// 1 for a mod pi = 0\
/ 2 \ || | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - \1 - tan (2*a)/*|<1 + cos(4*a) | + |< | + |< |
||------------ otherwise | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
\\ 2 /
$$\left(- \left(- \tan^{2}{\left(2 a \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cos{\left(4 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // / 3*pi\ \
|| | // 1 for a mod pi = 0\ || 1 for |4*a + ----| mod 2*pi = 0|
19 - |< 2 / 2 \ | + |< | + |< \ 2 / |
||sin (2*a)*\-1 + cot (2*a)/ otherwise | \\cos(2*a) otherwise / || |
\\ / \\sin(4*a) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 4 2 / 1 \ | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |<4*cos (a)*tan (a)*|-1 + ---------| otherwise | + |< | + |< |
|| | 2 | | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
|| \ tan (2*a)/ |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \cos^{4}{\left(a \right)} \tan^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + 19$$
4 4
4*sin (a) 4*sin (2*a)
1 - --------- 1 - -----------
2 2 2
sin (2*a) sin (4*a) 4*sin (2*a)
19 + ------------- - --------------- + --------------------------
4 4 / 4 \
4*sin (a) 4*sin (2*a) | 4*sin (2*a)|
1 + --------- 1 + ----------- |1 + -----------|*sin(4*a)
2 2 | 2 |
sin (2*a) sin (4*a) \ sin (4*a) /
$$\frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} - \frac{- \frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1}{\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1} + 19 + \frac{4 \sin^{2}{\left(2 a \right)}}{\left(\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \sin{\left(4 a \right)}}$$
// 1 for 2*a mod pi = 0\
|| |
|| / 2 \ | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |< 2 | sin (4*a) | | + |< | + |< |
||sin (2*a)*|-1 + -----------| otherwise | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
|| | 4 | |
\\ \ 4*sin (2*a)/ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| | // 1 for a mod pi = 0\
|| / 2 \ | // 0 for 4*a mod pi = 0\ || |
19 - |< 2 | sin (4*a) | | + |< | + |< /pi \ |
||sin (2*a)*|-1 + -----------| otherwise | \\sin(4*a) otherwise / ||sin|-- + 2*a| otherwise |
|| | 4 | | \\ \2 / /
\\ \ 4*sin (2*a)/ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| / 2 \ |
|| | sin (4*a) | | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
19 - |<(1 - cos(4*a))*|-1 + -----------| | + |< | + |< |
|| | 4 | | \\sin(4*a) otherwise / \\cos(2*a) otherwise /
|| \ 4*sin (2*a)/ |
||--------------------------------- otherwise |
\\ 2 /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \left(- \cos{\left(4 a \right)} + 1\right)}{2} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| / 2 \ | // 0 for 4*a mod pi = 0\
|| 2/ pi\ | cos (2*a) | | || | // 1 for a mod pi = 0\
19 - |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || | || |
|| 2 | || 2*cot(2*a) | || 2 |
19 - |<-1 + cot (2*a) | + |<------------- otherwise | + |<-1 + cot (a) |
||-------------- otherwise | || 2 | ||------------ otherwise |
|| 2 | ||1 + cot (2*a) | || 2 |
\\1 + cot (2*a) / \\ / \\1 + cot (a) /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 2 | // 1 for a mod pi = 0\
|| csc (2*a) | // 0 for 4*a mod pi = 0\ || |
||-1 + -------------- | || | || 1 |
19 - |< 2/pi \ | + |< 1 | + |<------------- otherwise |
|| csc |-- - 2*a| | ||-------- otherwise | || /pi \ |
|| \2 / | \\csc(4*a) / ||csc|-- - 2*a| |
||------------------- otherwise | \\ \2 / /
|| 2 |
\\ csc (2*a) /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 2/ pi\ |
|| sec |2*a - --| | // 0 for 4*a mod pi = 0\
|| \ 2 / | || | // 1 for a mod pi = 0\
||-1 + -------------- | || 1 | || |
19 - |< 2 | + |<------------- otherwise | + |< 1 |
|| sec (2*a) | || / pi\ | ||-------- otherwise |
||------------------- otherwise | ||sec|4*a - --| | \\sec(2*a) /
|| 2/ pi\ | \\ \ 2 / /
|| sec |2*a - --| |
|| \ 2 / |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 19$$
2/ pi\ 2/ pi\
cos |a - --| cos |2*a - --|
\ 2 / \ 2 /
1 - ------------ 1 - -------------- / pi\
2 2 2*cos|2*a - --|
cos (a) cos (2*a) \ 2 /
19 + ---------------- - ------------------ + -----------------------------
2/ pi\ 2/ pi\ / 2/ pi\\
cos |a - --| cos |2*a - --| | cos |2*a - --||
\ 2 / \ 2 / | \ 2 /|
1 + ------------ 1 + -------------- |1 + --------------|*cos(2*a)
2 2 | 2 |
cos (a) cos (2*a) \ cos (2*a) /
$$\frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} - \frac{1 - \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}}{1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}} + 19 + \frac{2 \cos{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \cos{\left(2 a \right)}}$$
2 2
sec (a) sec (2*a)
1 - ------------ 1 - --------------
2/ pi\ 2/ pi\
sec |a - --| sec |2*a - --|
\ 2 / \ 2 / 2*sec(2*a)
19 + ---------------- - ------------------ + ----------------------------------
2 2 / 2 \
sec (a) sec (2*a) | sec (2*a) | / pi\
1 + ------------ 1 + -------------- |1 + --------------|*sec|2*a - --|
2/ pi\ 2/ pi\ | 2/ pi\| \ 2 /
sec |a - --| sec |2*a - --| | sec |2*a - --||
\ 2 / \ 2 / \ \ 2 //
$$\frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} - \frac{- \frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1} + 19 + \frac{2 \sec{\left(2 a \right)}}{\left(\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
2/pi \ 2/pi \
csc |-- - a| csc |-- - 2*a|
\2 / \2 /
1 - ------------ 1 - -------------- /pi \
2 2 2*csc|-- - 2*a|
csc (a) csc (2*a) \2 /
19 + ---------------- - ------------------ + -----------------------------
2/pi \ 2/pi \ / 2/pi \\
csc |-- - a| csc |-- - 2*a| | csc |-- - 2*a||
\2 / \2 / | \2 /|
1 + ------------ 1 + -------------- |1 + --------------|*csc(2*a)
2 2 | 2 |
csc (a) csc (2*a) \ csc (2*a) /
$$\frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} - \frac{1 - \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}}{1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}} + 19 + \frac{2 \csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \csc{\left(2 a \right)}}$$
// /pi \ \
|| 0 for |-- + 4*a| mod pi = 0|
|| \2 / | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || | || |
|| / pi\ | || 2*cot(2*a) | || 2 |
19 - |< 2*cot|2*a + --| | + |<------------- otherwise | + |<-1 + cot (a) |
|| \ 4 / | || 2 | ||------------ otherwise |
||------------------ otherwise | ||1 + cot (2*a) | || 2 |
|| 2/ pi\ | \\ / \\1 + cot (a) /
||1 + cot |2*a + --| |
\\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ || |
|| 1 | || | || 1 |
||-1 + --------- | || 2 | ||-1 + ------- |
|| 2 | ||------------------------ otherwise | || 2 |
19 - |< tan (2*a) | + |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{1 + \frac{1}{\tan^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| |
|| 2 / 1 \ | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
||4*tan (a)*|-1 + ---------| | || | || |
|| | 2 | | || 2*tan(2*a) | || 2 |
19 - |< \ tan (2*a)/ | + |<------------- otherwise | + |<1 - tan (a) |
||-------------------------- otherwise | || 2 | ||----------- otherwise |
|| 2 | ||1 + tan (2*a) | || 2 |
|| / 2 \ | \\ / \\1 + tan (a) /
|| \1 + tan (a)/ |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 19$$
// / 3*pi\ \
|| 1 for |4*a + ----| mod 2*pi = 0|
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ || \ 2 / |
|| | || | || |
|| 2 | || 2 | || 2/ pi\ |
19 - |<-1 + cot (2*a) | + |<-1 + cot (a) | + |<-1 + tan |2*a + --| |
||-------------- otherwise | ||------------ otherwise | || \ 4 / |
|| 2 | || 2 | ||------------------- otherwise |
\\1 + cot (2*a) / \\1 + cot (a) / || 2/ pi\ |
|| 1 + tan |2*a + --| |
\\ \ 4 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| 2 | || | || |
/ 1 \ ||/ 2 \ | || 2*cot(2*a) | || 2 |
19 - |1 - ---------|*|<\-1 + cot (a)/ | + |<------------- otherwise | + |<-1 + cot (a) |
| 2 | ||--------------- otherwise | || 2 | ||------------ otherwise |
\ cot (2*a)/ || 2 | ||1 + cot (2*a) | || 2 |
|| / 2 \ | \\ / \\1 + cot (a) /
\\ \1 + cot (a)/ /
$$\left(- \left(1 - \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(a \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 4*a| mod pi = 0| || 0 for |-- + 2*a| mod pi = 0|
|| \2 / | // 0 for 4*a mod pi = 0\ || \2 / |
|| | || | || |
|| / pi\ | || 2*cot(2*a) | || / pi\ |
19 - |< 2*cot|2*a + --| | + |<------------- otherwise | + |< 2*cot|a + --| |
|| \ 4 / | || 2 | || \ 4 / |
||------------------ otherwise | ||1 + cot (2*a) | ||---------------- otherwise |
|| 2/ pi\ | \\ / || 2/ pi\ |
||1 + cot |2*a + --| | ||1 + cot |a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| // 0 for 2*a mod pi = 0\ | || | || |
19 - |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{- \cos{\left(4 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| sin (4*a) | || | || sin (2*a) |
||-1 + ----------- | || sin(4*a) | ||-1 + --------- |
|| 4 | ||--------------------------- otherwise | || 4 |
19 - |< 4*sin (2*a) | + |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{\sin{\left(4 a \right)}}{\left(1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}}{1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| cos (2*a) | || | || cos (a) |
||-1 + -------------- | || 2*cos(2*a) | ||-1 + ------------ |
|| 2/ pi\ | ||---------------------------------- otherwise | || 2/ pi\ |
|| cos |2*a - --| | ||/ 2 \ | || cos |a - --| |
19 - |< \ 2 / | + |<| cos (2*a) | / pi\ | + |< \ 2 / |
||------------------- otherwise | |||1 + --------------|*cos|2*a - --| | ||----------------- otherwise |
|| 2 | ||| 2/ pi\| \ 2 / | || 2 |
|| cos (2*a) | ||| cos |2*a - --|| | || cos (a) |
|| 1 + -------------- | ||\ \ 2 // | || 1 + ------------ |
|| 2/ pi\ | \\ / || 2/ pi\ |
|| cos |2*a - --| | || cos |a - --| |
\\ \ 2 / / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cos{\left(2 a \right)}}{\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ || |
|| 2/ pi\ | || | || 2/ pi\ |
|| sec |2*a - --| | || / pi\ | || sec |a - --| |
|| \ 2 / | || 2*sec|2*a - --| | || \ 2 / |
||-1 + -------------- | || \ 2 / | ||-1 + ------------ |
|| 2 | ||----------------------------- otherwise | || 2 |
19 - |< sec (2*a) | + |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}\right) \sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\
|| | // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| // 0 for 2*a mod pi = 0\ | || | || |
|| || | | ||/ 0 for 4*a mod pi = 0 | ||/ 1 for a mod pi = 0 |
|| || 2 | | ||| | ||| |
19 - |
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(a \right)}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + 19$$
// 1 for 2*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| csc (2*a) | || | || csc (a) |
||-1 + -------------- | || 2*csc(2*a) | ||-1 + ------------ |
|| 2/pi \ | ||---------------------------------- otherwise | || 2/pi \ |
|| csc |-- - 2*a| | ||/ 2 \ | || csc |-- - a| |
19 - |< \2 / | + |<| csc (2*a) | /pi \ | + |< \2 / |
||------------------- otherwise | |||1 + --------------|*csc|-- - 2*a| | ||----------------- otherwise |
|| 2 | ||| 2/pi \| \2 / | || 2 |
|| csc (2*a) | ||| csc |-- - 2*a|| | || csc (a) |
|| 1 + -------------- | ||\ \2 // | || 1 + ------------ |
|| 2/pi \ | \\ / || 2/pi \ |
|| csc |-- - 2*a| | || csc |-- - a| |
\\ \2 / / \\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \csc{\left(2 a \right)}}{\left(\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 19$$
19 - Piecewise((1, Mod(2*a = pi, 0)), ((-1 + csc(2*a)^2/csc(pi/2 - 2*a)^2)/(1 + csc(2*a)^2/csc(pi/2 - 2*a)^2), True)) + Piecewise((0, Mod(4*a = pi, 0)), (2*csc(2*a)/((1 + csc(2*a)^2/csc(pi/2 - 2*a)^2)*csc(pi/2 - 2*a)), True)) + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))