Господин Экзамен
Общий знаменатель sin(a)/1+cos(a)+1+cos(a)/sin(a)
Решение
Вы ввели
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sin(a) cos(a) ------ + cos(a) + 1 + ------ 1 sin(a)
$$\frac{\sin{\left(a \right)}}{1} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
sin(a)/1 + cos(a) + 1 + cos(a)/sin(a)
Общее упрощение
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___ / 1 \ / pi\
\/ 2 *|1 + ------|*sin|a + --|
\ sin(a)/ \ 4 /
$$\sqrt{2} \cdot \left(1 + \frac{1}{\sin{\left(a \right)}}\right) \sin{\left(a + \frac{\pi}{4} \right)}$$
sqrt(2)*(1 + 1/sin(a))*sin(a + pi/4)
Собрать выражение
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1 + cos(a) + cot(a) + sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + \cot{\left(a \right)} + 1$$
cos(a)
1 + ------ + cos(a) + sin(a)
sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
1 + cos(a)/sin(a) + cos(a) + sin(a)
Объединение рациональных выражений
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2
sin (a) + cos(a)*sin(a) + cos(a) + sin(a)
-----------------------------------------
sin(a)
$$\frac{\sin^{2}{\left(a \right)} + \sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)} + \cos{\left(a \right)}}{\sin{\left(a \right)}}$$
(sin(a)^2 + cos(a)*sin(a) + cos(a) + sin(a))/sin(a)
Тригонометрическая часть
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1 ___ / pi\
1 + ------ + \/ 2 *cos|a - --|
tan(a) \ 4 /
$$\sqrt{2} \cos{\left(a - \frac{\pi}{4} \right)} + 1 + \frac{1}{\tan{\left(a \right)}}$$
1 ___ / pi\
1 + ------ + \/ 2 *sin|a + --|
tan(a) \ 4 /
$$\sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} + 1 + \frac{1}{\tan{\left(a \right)}}$$
cos(a)
1 + ------ + cos(a) + sin(a)
sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
___ / pi\ sin(2*a)
1 + \/ 2 *sin|a + --| + ---------
\ 4 / 2
2*sin (a)
$$\sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} + 1 + \frac{\sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}}$$
___ / pi\ cos(a)
1 + \/ 2 *cos|a - --| + -----------
\ 4 / / pi\
cos|a - --|
\ 2 /
$$\sqrt{2} \cos{\left(a - \frac{\pi}{4} \right)} + \frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + 1$$
1 1 csc(a)
1 + ------ + ------ + ------
csc(a) sec(a) sec(a)
$$\frac{\csc{\left(a \right)}}{\sec{\left(a \right)}} + 1 + \frac{1}{\sec{\left(a \right)}} + \frac{1}{\csc{\left(a \right)}}$$
/ pi\
___ sec|a - --|
\/ 2 \ 2 /
1 + ----------- + -----------
/ pi\ sec(a)
sec|a - --|
\ 4 /
$$1 + \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{\sqrt{2}}{\sec{\left(a - \frac{\pi}{4} \right)}}$$
___
\/ 2 csc(a)
1 + ----------- + -----------
/ pi\ /pi \
csc|a + --| csc|-- - a|
\ 4 / \2 /
$$\frac{\csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + 1 + \frac{\sqrt{2}}{\csc{\left(a + \frac{\pi}{4} \right)}}$$
/ pi\
sin|a + --|
\ 2 / / pi\
1 + ----------- + sin(a) + sin|a + --|
sin(a) \ 2 /
$$\sin{\left(a \right)} + \sin{\left(a + \frac{\pi}{2} \right)} + 1 + \frac{\sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}}$$
cos(a) / pi\
1 + ----------- + cos(a) + cos|a - --|
/ pi\ \ 2 /
cos|a - --|
\ 2 /
$$\cos{\left(a \right)} + \cos{\left(a - \frac{\pi}{2} \right)} + \frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + 1$$
/a\
tan|-|
1 \2/ ___ / pi\
1 + -------- - ------ + \/ 2 *cos|a - --|
/a\ 2 \ 4 /
2*tan|-|
\2/
$$\sqrt{2} \cos{\left(a - \frac{\pi}{4} \right)} - \frac{\tan{\left(\frac{a}{2} \right)}}{2} + 1 + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}}$$
/a\
tan|-|
1 \2/ ___ / pi\
1 + -------- - ------ + \/ 2 *sin|a + --|
/a\ 2 \ 4 /
2*tan|-|
\2/
$$\sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} - \frac{\tan{\left(\frac{a}{2} \right)}}{2} + 1 + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}}$$
/ pi\
sec|a - --|
1 1 \ 2 /
1 + ------ + ----------- + -----------
sec(a) / pi\ sec(a)
sec|a - --|
\ 2 /
$$1 + \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a \right)}}$$
/pi \
sec|-- - a|
1 1 \2 /
1 + ------ + ----------- + -----------
sec(a) /pi \ sec(a)
sec|-- - a|
\2 /
$$1 + \frac{\sec{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a \right)}}$$
1 1 csc(a)
1 + ------ + ----------- + -----------
csc(a) /pi \ /pi \
csc|-- - a| csc|-- - a|
\2 / \2 /
$$\frac{\csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + 1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(a \right)}}$$
___ /a pi\
2*\/ 2 *tan|- + --|
1 \2 8 /
1 + ------ + -------------------
tan(a) 2/a pi\
1 + tan |- + --|
\2 8 /
$$1 + \frac{2 \sqrt{2} \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{1}{\tan{\left(a \right)}}$$
2/a\
2*cos |-|
1 2/a\ \2/
- ------ + 2*cos |-| + --------- + sin(a)
sin(a) \2/ sin(a)
$$2 \cos^{2}{\left(\frac{a}{2} \right)} + \sin{\left(a \right)} + \frac{2 \cos^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \frac{1}{\sin{\left(a \right)}}$$
1 1 csc(pi - a)
1 + ----------- + ----------- + -----------
csc(pi - a) /pi \ /pi \
csc|-- - a| csc|-- - a|
\2 / \2 /
$$\frac{\csc{\left(- a + \pi \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + 1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- a + \pi \right)}}$$
2/a\
sin |-|
___ / pi\ \2/ sin(a)
1 + \/ 2 *sin|a + --| - ------- + ---------
\ 4 / sin(a) 2/a\
4*sin |-|
\2/
$$- \frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} + 1 + \frac{\sin{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)}}$$
/a\ cos(a)
1 + (1 + cos(a))*tan|-| + ------------------- + cos(a)
\2/ /a\
(1 + cos(a))*tan|-|
\2/
$$\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
2/a\
sin |-|
___ / pi\ sin(a) \2/
1 + \/ 2 *sin|a + --| + -------------- - -------
\ 4 / 2*(1 - cos(a)) sin(a)
$$- \frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \sqrt{2} \sin{\left(a + \frac{\pi}{4} \right)} + 1 + \frac{\sin{\left(a \right)}}{2 \cdot \left(- \cos{\left(a \right)} + 1\right)}$$
/a\ ___ /a pi\
tan|-| 2*\/ 2 *tan|- + --|
1 \2/ \2 8 /
1 + -------- - ------ + -------------------
/a\ 2 2/a pi\
2*tan|-| 1 + tan |- + --|
\2/ \2 8 /
$$- \frac{\tan{\left(\frac{a}{2} \right)}}{2} + 1 + \frac{2 \sqrt{2} \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}}$$
/a\ /a pi\
cos|-| cos|- - --|
___ / pi\ \2/ \2 2 /
1 + \/ 2 *cos|a - --| + ------------- - -----------
\ 4 / /a pi\ /a\
2*cos|- - --| 2*cos|-|
\2 2 / \2/
$$\sqrt{2} \cos{\left(a - \frac{\pi}{4} \right)} + \frac{\cos{\left(\frac{a}{2} \right)}}{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1 - \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}}$$
1 /a\
------ - tan|-|
/a\ \2/ / 2/a pi\\
tan|-| |1 - cot |- + --||*(1 + sin(a))
\2/ \ \2 4 //
1 + --------------- + ------------------------------- + cos(a)
2 2
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + \frac{- \tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}}{2} + \cos{\left(a \right)} + 1$$
/a pi\ /a\
___ sec|- - --| sec|-|
\/ 2 \2 2 / \2/
1 + ----------- + ----------- - -------------
/ pi\ /a\ /a pi\
sec|a - --| 2*sec|-| 2*sec|- - --|
\ 4 / \2/ \2 2 /
$$- \frac{\sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{a}{2} \right)}} + \frac{\sqrt{2}}{\sec{\left(a - \frac{\pi}{4} \right)}}$$
/a\ /pi a\
___ csc|-| csc|-- - -|
\/ 2 \2/ \2 2/
1 + ----------- + ------------- - -----------
/ pi\ /pi a\ /a\
csc|a + --| 2*csc|-- - -| 2*csc|-|
\ 4 / \2 2/ \2/
$$\frac{\csc{\left(\frac{a}{2} \right)}}{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1 - \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} + \frac{\sqrt{2}}{\csc{\left(a + \frac{\pi}{4} \right)}}$$
2/a\ 2/a\ /a\
1 - tan |-| 1 - tan |-| 2*tan|-|
\2/ \2/ \2/
1 + ----------- + ----------- + -----------
2/a\ /a\ 2/a\
1 + tan |-| 2*tan|-| 1 + tan |-|
\2/ \2/ \2/
$$\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{2 \tan{\left(\frac{a}{2} \right)}} + \frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
___ || \ 4 / |
1 + \/ 2 *|< | + cot(a)
|| 2/a pi\ /a pi\ |
||2*sin |- + --|*cot|- + --| otherwise |
\\ \2 8 / \2 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + \cot{\left(a \right)} + 1$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
|| \ 4 / |
|| |
___ || /a pi\ |
1 + \/ 2 *|< 2*cot|- + --| | + cot(a)
|| \2 8 / |
||---------------- otherwise |
|| 2/a pi\ |
||1 + cot |- + --| |
\\ \2 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \cot{\left(a \right)} + 1$$
1 / 1 \ /a\
1 - ------- |1 - -------|*cot|-|
2/a\ | 2/a\| \2/
cot |-| | cot |-||
\2/ \ \2// 2
1 + ----------- + -------------------- + --------------------
1 2 / 1 \ /a\
1 + ------- |1 + -------|*cot|-|
2/a\ | 2/a\| \2/
cot |-| | cot |-||
\2/ \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}{2} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 1 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}$$
/a\ /a\ // / pi\ \
cot|-| tan|-| || 0 for |a + --| mod pi = 0|
\2/ \2/ ___ || \ 4 / |
1 + ------ - ------ + \/ 2 *|< |
2 2 || 2/a pi\ /a pi\ |
||2*sin |- + --|*cot|- + --| otherwise |
\\ \2 8 / \2 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) - \frac{\tan{\left(\frac{a}{2} \right)}}{2} + \frac{\cot{\left(\frac{a}{2} \right)}}{2} + 1$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
/a\ || \ 4 / |
cot|-| || |
\2/ 1 ___ || /a pi\ |
1 + ------ - -------- + \/ 2 *|< 2*cot|- + --| |
2 /a\ || \2 8 / |
2*cot|-| ||---------------- otherwise |
\2/ || 2/a pi\ |
||1 + cot |- + --| |
\\ \2 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \frac{\cot{\left(\frac{a}{2} \right)}}{2} + 1 - \frac{1}{2 \cot{\left(\frac{a}{2} \right)}}$$
/a\ /a pi\ / 2/a\\ /a pi\
2*cot|-| 2*tan|- + --| |1 + cot |-||*tan|- + --|
\2/ \2 4 / \ \2// \2 4 /
1 + ----------- + ---------------- + -------------------------
2/a\ 2/a pi\ / 2/a pi\\ /a\
1 + cot |-| 1 + tan |- + --| |1 + tan |- + --||*cot|-|
\2/ \2 4 / \ \2 4 // \2/
$$1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}$$
/a\ /a pi\ / 2/a\\ /a pi\
2*tan|-| 2*tan|- + --| |1 + tan |-||*tan|- + --|
\2/ \2 4 / \ \2// \2 4 /
1 + ----------- + ---------------- + -------------------------
2/a\ 2/a pi\ / 2/a pi\\ /a\
1 + tan |-| 1 + tan |- + --| |1 + tan |- + --||*tan|-|
\2/ \2 4 / \ \2 4 // \2/
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}} + 1 + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
2/a\
4*sin |-|*sin(a)
1 - 2*cos(a) + cos(2*a) 2*(-1 - cos(2*a) + 2*cos(a)) \2/
1 + ----------------------- + ------------------------------ + -------------------
-2*sin(a) + sin(2*a) 2 2 4/a\
1 - cos(2*a) + 2*(1 - cos(a)) sin (a) + 4*sin |-|
\2/
$$\frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + 1 + \frac{2 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} + \frac{- 2 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 1}{- 2 \sin{\left(a \right)} + \sin{\left(2 a \right)}}$$
2/a\ 2/a pi\ / 2/a pi\\ / 2/a\\
-1 + cot |-| -1 + tan |- + --| |1 + tan |- + --||*|-1 + cot |-||
\2/ \2 4 / \ \2 4 // \ \2//
1 + ------------ + ----------------- + ---------------------------------
2/a\ 2/a pi\ / 2/a\\ / 2/a pi\\
1 + cot |-| 1 + tan |- + --| |1 + cot |-||*|-1 + tan |- + --||
\2/ \2 4 / \ \2// \ \2 4 //
$$\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1 + \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2/a pi\ 2/a\ / 2/a pi\\ / 2/a\\
1 - cot |- + --| 1 - tan |-| |1 + cot |- + --||*|1 - tan |-||
\2 4 / \2/ \ \2 4 // \ \2//
1 + ---------------- + ----------- + --------------------------------
2/a pi\ 2/a\ / 2/a\\ / 2/a pi\\
1 + cot |- + --| 1 + tan |-| |1 + tan |-||*|1 - cot |- + --||
\2 4 / \2/ \ \2// \ \2 4 //
$$\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} + \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1$$
4/a\ / 4/a\\
4*sin |-| | 4*sin |-||
\2/ | \2/|
1 - --------- 2/a\ |1 - ---------|*sin(a)
2 4*sin |-| | 2 |
sin (a) \2/ \ sin (a) /
1 + ------------- + ---------------------- + ----------------------
4/a\ / 4/a\\ 2/a\
4*sin |-| | 4*sin |-|| 4*sin |-|
\2/ | \2/| \2/
1 + --------- |1 + ---------|*sin(a)
2 | 2 |
sin (a) \ sin (a) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)}} + \frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + 1 + \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< |*|< | + |< | + |< |
\\cos(a) otherwise / \\csc(a) otherwise / \\sin(a) otherwise / \\cos(a) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\csc{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
// zoo for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< |*|< 1 | + |< | + |< |
\\cos(a) otherwise / ||------ otherwise | \\sin(a) otherwise / \\cos(a) otherwise /
\\sin(a) /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
// zoo for a mod pi = 0\
|| | // 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || 1 | || | // 1 for a mod 2*pi = 0\
1 + |< |*|<----------- otherwise | + |< / pi\ | + |< |
\\cos(a) otherwise / || / pi\ | ||cos|a - --| otherwise | \\cos(a) otherwise /
||cos|a - --| | \\ \ 2 / /
\\ \ 2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || | // 0 for a mod pi = 0\ || |
1 + |< / pi\ |*|< 1 | + |< | + |< / pi\ |
||sin|a + --| otherwise | ||------ otherwise | \\sin(a) otherwise / ||sin|a + --| otherwise |
\\ \ 2 / / \\sin(a) / \\ \ 2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ || | // 1 for a mod 2*pi = 0\
|| | || | || 1 | || |
1 + |< 1 |*|< / pi\ | + |<----------- otherwise | + |< 1 |
||------ otherwise | ||sec|a - --| otherwise | || / pi\ | ||------ otherwise |
\\sec(a) / \\ \ 2 / / ||sec|a - --| | \\sec(a) /
\\ \ 2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\sec{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\ || |
|| 1 | // zoo for a mod pi = 0\ || | || 1 |
1 + |<----------- otherwise |*|< | + |< 1 | + |<----------- otherwise |
|| /pi \ | \\csc(a) otherwise / ||------ otherwise | || /pi \ |
||csc|-- - a| | \\csc(a) / ||csc|-- - a| |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\csc{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// zoo for a mod pi = 0\ // 0 for a mod pi = 0\
|| | || |
// 1 for a mod 2*pi = 0\ || /a\ | ||1 - cos(a) | // 1 for a mod 2*pi = 0\
1 + |< |*|< -tan|-| | + |<---------- otherwise | + |< |
\\cos(a) otherwise / || \2/ | || /a\ | \\cos(a) otherwise /
||----------- otherwise | || tan|-| |
\\-1 + cos(a) / \\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\- \frac{\tan{\left(\frac{a}{2} \right)}}{\cos{\left(a \right)} - 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
|| | // / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 /a\ | // 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
1 + |< |*|<------ + tan|-| | + |< | + |< \ 2 / |
\\cos(a) otherwise / || /a\ \2/ | \\cos(a) otherwise / || |
||tan|-| | \\sin(a) otherwise /
|| \2/ |
||--------------- otherwise |
\\ 2 /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}}{2} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
2/a pi\ / 2/a pi\\
cos |- - --| | cos |- - --||
\2 2 / | \2 2 /| /a\
1 - ------------ |1 - ------------|*cos|-|
2/a\ | 2/a\ | \2/ /a pi\
cos |-| | cos |-| | 2*cos|- - --|
\2/ \ \2/ / \2 2 /
1 + ---------------- + ------------------------- + -------------------------
2/a pi\ /a pi\ / 2/a pi\\
cos |- - --| 2*cos|- - --| | cos |- - --||
\2 2 / \2 2 / | \2 2 /| /a\
1 + ------------ |1 + ------------|*cos|-|
2/a\ | 2/a\ | \2/
cos |-| | cos |-| |
\2/ \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + 1 + \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}$$
2/a\ / 2/a\ \
sec |-| | sec |-| |
\2/ | \2/ | /a pi\
1 - ------------ |1 - ------------|*sec|- - --|
2/a pi\ | 2/a pi\| \2 2 / /a\
sec |- - --| | sec |- - --|| 2*sec|-|
\2 2 / \ \2 2 // \2/
1 + ---------------- + ------------------------------ + ------------------------------
2/a\ /a\ / 2/a\ \
sec |-| 2*sec|-| | sec |-| |
\2/ \2/ | \2/ | /a pi\
1 + ------------ |1 + ------------|*sec|- - --|
2/a pi\ | 2/a pi\| \2 2 /
sec |- - --| | sec |- - --||
\2 2 / \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{a}{2} \right)}} + \frac{- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + 1 + \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
2/pi a\ / 2/pi a\\
csc |-- - -| | csc |-- - -||
\2 2/ | \2 2/| /a\
1 - ------------ |1 - ------------|*csc|-|
2/a\ | 2/a\ | \2/ /pi a\
csc |-| | csc |-| | 2*csc|-- - -|
\2/ \ \2/ / \2 2/
1 + ---------------- + ------------------------- + -------------------------
2/pi a\ /pi a\ / 2/pi a\\
csc |-- - -| 2*csc|-- - -| | csc |-- - -||
\2 2/ \2 2/ | \2 2/| /a\
1 + ------------ |1 + ------------|*csc|-|
2/a\ | 2/a\ | \2/
csc |-| | csc |-| |
\2/ \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + 1 + \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}$$
// / pi\ \ // / pi\ \
|| 0 for |a + --| mod pi = 0| // zoo for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| \ 2 / | || | // 0 for a mod pi = 0\ || \ 2 / |
1 + |< |*|< 1 | + |< | + |< |
|| /a pi\ | ||------ otherwise | \\sin(a) otherwise / || /a pi\ |
||(1 + sin(a))*cot|- + --| otherwise | \\sin(a) / ||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || | || | || |
|| 2/a\ | || 2/a\ | || /a\ | || 2/a\ |
||-1 + cot |-| | ||1 + cot |-| | || 2*cot|-| | ||-1 + cot |-| |
1 + |< \2/ |*|< \2/ | + |< \2/ | + |< \2/ |
||------------ otherwise | ||----------- otherwise | ||----------- otherwise | ||------------ otherwise |
|| 2/a\ | || /a\ | || 2/a\ | || 2/a\ |
||1 + cot |-| | || 2*cot|-| | ||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || | || | || |
|| 2/a\ | || 2/a\ | || /a\ | || 2/a\ |
||1 - tan |-| | ||1 + tan |-| | || 2*tan|-| | ||1 - tan |-| |
1 + |< \2/ |*|< \2/ | + |< \2/ | + |< \2/ |
||----------- otherwise | ||----------- otherwise | ||----------- otherwise | ||----------- otherwise |
|| 2/a\ | || /a\ | || 2/a\ | || 2/a\ |
||1 + tan |-| | || 2*tan|-| | ||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 1 | // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 1 |
||-1 + ------- | || | || | ||-1 + ------- |
|| 2/a\ | ||/ 1 \ /a\ | || 2 | || 2/a\ |
|| tan |-| | |||1 + -------|*tan|-| | ||-------------------- otherwise | || tan |-| |
1 + |< \2/ |*|<| 2/a\| \2/ | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{2} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1$$
// zoo for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | ||/ zoo for a mod pi = 0 | || | || |
1 + |
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// / pi\ \ // / pi\ \
|| 0 for |a + --| mod pi = 0| // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| \ 2 / | || | || | || \ 2 / |
|| | || 2/a\ | || /a\ | || |
|| /a pi\ | ||1 + cot |-| | || 2*cot|-| | || /a pi\ |
1 + |< 2*cot|- + --| |*|< \2/ | + |< \2/ | + |< 2*cot|- + --| |
|| \2 4 / | ||----------- otherwise | ||----------- otherwise | || \2 4 / |
||---------------- otherwise | || /a\ | || 2/a\ | ||---------------- otherwise |
|| 2/a pi\ | || 2*cot|-| | ||1 + cot |-| | || 2/a pi\ |
||1 + cot |- + --| | \\ \2/ / \\ \2/ / ||1 + cot |- + --| |
\\ \2 4 / / \\ \2 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// / 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 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2/a\ | || | || 2/a\ | || |
||-1 + cot |-| | || 2/a pi\ | ||-1 + cot |-| | || 2/a pi\ |
1 + |< \2/ |*|< 1 + tan |- + --| | + |< \2/ | + |<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / | ||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise | || 2/a\ | ||----------------- otherwise |
||1 + cot |-| | || 2/a pi\ | ||1 + cot |-| | || 2/a pi\ |
\\ \2/ / ||-1 + tan |- + --| | \\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / / \\ \2 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
// zoo for a mod pi = 0\ || |
// 1 for a mod 2*pi = 0\ || | || 2*sin(a) | // 1 for a mod 2*pi = 0\
|| | || 2/a\ | ||---------------------------- otherwise | || |
|| 2 | || sin |-| | || / 2 \ | || 2 |
1 + |< -4 + 4*sin (a) + 4*cos(a) |*|< 1 \2/ | + |< | sin (a) | | + |< -4 + 4*sin (a) + 4*cos(a) |
||--------------------------- otherwise | ||-------- + ------- otherwise | ||(1 - cos(a))*|1 + ---------| | ||--------------------------- otherwise |
|| 2 2 | || /a\ sin(a) | || | 4/a\| | || 2 2 |
\\2*(1 - cos(a)) + 2*sin (a) / ||2*tan|-| | || | 4*sin |-|| | \\2*(1 - cos(a)) + 2*sin (a) /
\\ \2/ / || \ \2// |
\\ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2 | // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 2 |
|| sin (a) | || | || | || sin (a) |
||-1 + --------- | || / 2 \ | || sin(a) | ||-1 + --------- |
|| 4/a\ | || 2/a\ | sin (a) | | ||----------------------- otherwise | || 4/a\ |
|| 4*sin |-| | ||sin |-|*|1 + ---------| | ||/ 2 \ | || 4*sin |-| |
1 + |< \2/ |*|< \2/ | 4/a\| | + |<| sin (a) | 2/a\ | + |< \2/ |
||-------------- otherwise | || | 4*sin |-|| | |||1 + ---------|*sin |-| | ||-------------- otherwise |
|| 2 | || \ \2// | ||| 4/a\| \2/ | || 2 |
|| sin (a) | ||----------------------- otherwise | ||| 4*sin |-|| | || sin (a) |
||1 + --------- | || sin(a) | ||\ \2// | ||1 + --------- |
|| 4/a\ | \\ / \\ / || 4/a\ |
|| 4*sin |-| | || 4*sin |-| |
\\ \2/ / \\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || | || | || |
||/ 1 for a mod 2*pi = 0 | ||/ zoo for a mod pi = 0 | ||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 |
||| | ||| | ||| | ||| |
||| 2/a\ | ||| 2/a\ | ||| /a\ | ||| 2/a\ |
1 + |<|-1 + cot |-| |*|<|1 + cot |-| | + |<| 2*cot|-| | + |<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||----------- otherwise | |||----------- otherwise | |||------------ otherwise |
||| 2/a\ | ||| /a\ | ||| 2/a\ | ||| 2/a\ |
|||1 + cot |-| | ||| 2*cot|-| | |||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ / \\\ \2/ / \\\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 2/a\ |
|| cos |-| | || | || | || cos |-| |
|| \2/ | ||/ 2/a\ \ | || /a\ | || \2/ |
||-1 + ------------ | ||| cos |-| | | || 2*cos|-| | ||-1 + ------------ |
|| 2/a pi\ | ||| \2/ | /a pi\ | || \2/ | || 2/a pi\ |
|| cos |- - --| | |||1 + ------------|*cos|- - --| | ||------------------------------ otherwise | || cos |- - --| |
1 + |< \2 2 / |*|<| 2/a pi\| \2 2 / | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a pi\ | // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 2/a pi\ |
|| sec |- - --| | || | || | || sec |- - --| |
|| \2 2 / | ||/ 2/a pi\\ | || /a pi\ | || \2 2 / |
||-1 + ------------ | ||| sec |- - --|| | || 2*sec|- - --| | ||-1 + ------------ |
|| 2/a\ | ||| \2 2 /| /a\ | || \2 2 / | || 2/a\ |
|| sec |-| | |||1 + ------------|*sec|-| | ||------------------------- otherwise | || sec |-| |
1 + |< \2/ |*|<| 2/a\ | \2/ | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | // zoo for a mod pi = 0\ // 0 for a mod pi = 0\ || 2/a\ |
|| csc |-| | || | || | || csc |-| |
|| \2/ | ||/ 2/a\ \ | || /a\ | || \2/ |
||-1 + ------------ | ||| csc |-| | | || 2*csc|-| | ||-1 + ------------ |
|| 2/pi a\ | ||| \2/ | /pi a\ | || \2/ | || 2/pi a\ |
|| csc |-- - -| | |||1 + ------------|*csc|-- - -| | ||------------------------------ otherwise | || csc |-- - -| |
1 + |< \2 2/ |*|<| 2/pi a\| \2 2/ | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1$$
1 + Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))*Piecewise((±oo, Mod(a = pi, 0)), ((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)/(2*csc(a/2)), True)) + Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True)) + Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))
Рациональный знаменатель
[src]
cos(a)
1 + ------ + cos(a) + sin(a)
sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
(1 + cos(a) + sin(a))*sin(a) + cos(a)
-------------------------------------
sin(a)
$$\frac{\left(\sin{\left(a \right)} + \cos{\left(a \right)} + 1\right) \sin{\left(a \right)} + \cos{\left(a \right)}}{\sin{\left(a \right)}}$$
((1 + cos(a) + sin(a))*sin(a) + cos(a))/sin(a)
Численный ответ
[src]
1.0 + 1.0*sin(a) + cos(a)/sin(a) + cos(a)
1.0 + 1.0*sin(a) + cos(a)/sin(a) + cos(a)
Общий знаменатель
[src]
cos(a)
1 + ------ + cos(a) + sin(a)
sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
1 + cos(a)/sin(a) + cos(a) + sin(a)
Степени
[src]
cos(a)
1 + ------ + cos(a) + sin(a)
sin(a)
$$\sin{\left(a \right)} + \cos{\left(a \right)} + 1 + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}}$$
/ I*a -I*a\
|e e |
I*a -I*a / -I*a I*a\ 2*I*|---- + -----|
e e I*\- e + e / \ 2 2 /
1 + ---- + ----- - ------------------ + ------------------
2 2 2 -I*a I*a
- e + e
$$- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} + \frac{e^{i a}}{2} + \frac{2 i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{e^{i a} - e^{- i a}} + 1 + \frac{e^{- i a}}{2}$$
1 + exp(i*a)/2 + exp(-i*a)/2 - i*(-exp(-i*a) + exp(i*a))/2 + 2*i*(exp(i*a)/2 + exp(-i*a)/2)/(-exp(-i*a) + exp(i*a))
Комбинаторика
[src]
(1 + sin(a))*(cos(a) + sin(a))
------------------------------
sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \left(\sin{\left(a \right)} + \cos{\left(a \right)}\right)}{\sin{\left(a \right)}}$$
(1 + sin(a))*(cos(a) + sin(a))/sin(a)