Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 21*a^3+8*a^2*b-5*a*b^2+2*b-6*a если a=1/2
- (x^7-22*x^5-16*x^4+129*x^3+160*x^2-108*x-144)/(x^4+2*x^3-13*x^2-38*x-2424) если x=2
- cos(2*a)-cos(4*a)+sin(4*a) если a=-1/3
- sin(2*l)+(sin(l)-cos(l))*2 если l=1/3
- Разложить многочлен на множители:
- m^6-27^2
- 72*x^4+23*x^2-95
- 25*x^2-10*x*y+y^2-9
- 3*y^2-27*y
- Общий знаменатель:
- (sin(pi/4+x)-cos(pi/4+x))/(sin(pi/4+x)+cos(pi/4+x))
- 256*(atan((2*x+4)/4)/16+(x+2)/(64+32*x+8*x^2))
- x-12*a/x^2-16*a^2-4*a/4*a*x-x^2
- (z-2)/(6*z+(z-2)*(z-2))+(6/(z*z*z-8))
- Умножение столбиком:
- 27*65
- 60*53
- 92*50
- 93*73
- Деление столбиком:
- 54654/4
- 31806/6
- 908/7
- 489/8
- Раскрыть скобки в:
- (5*d-8)*(8+5*d)
- (x-4)*(x-7)*(x-10)*(x-13)
- 25*b*(b-1)-(5*b+2)^2
- 8*p+(2-3*p)*(-1)-(5*p-2)
- Разложение числа на простые множители:
- 2104
- 4306
- 2296
- 2546
- Производная:
- 1/3
- График функции y =:
- 1/3
- Интеграл d{x}:
-
1/3
-
Идентичные выражения
- sin(два *l)+(sin(l)-cos(l))* два если l= один / три
- синус от (2 умножить на l) плюс ( синус от (l) минус косинус от (l)) умножить на 2 если l равно 1 делить на 3
- синус от (два умножить на l) плюс ( синус от (l) минус косинус от (l)) умножить на два если l равно один делить на три
- sin(2l)+(sin(l)-cos(l))2 если l=1/3
- sin2l+sinl-cosl2 если l=1/3
- sin(2*l)+(sin(l)-cos(l))*2 при l=1/3
- sin(2*l)+(sin(l)-cos(l))*2 если l=1 разделить на 3
-
Похожие выражения
- sin(2*l)-(sin(l)-cos(l))*2 если l=1/3
- sin(2*l)+(sin(l)+cos(l))*2 если l=1/3
-
Что Вы имели ввиду?
- sin(2*l) + (sin(l) - cos(l))^2
sin(2*l)+(sin(l)-cos(l))*2 если l=1/3
Решение
Вы ввели
[src]
sin(2*l) + (sin(l) - cos(l))*2
$$\left(\sin{\left(l \right)} - \cos{\left(l \right)}\right) 2 + \sin{\left(2 l \right)}$$
sin(2*l) + (sin(l) - cos(l))*2
Общее упрощение
[src]
___ / pi\
- 2*\/ 2 *cos|l + --| + sin(2*l)
\ 4 /
$$\sin{\left(2 l \right)} - 2 \sqrt{2} \cos{\left(l + \frac{\pi}{4} \right)}$$
-2*sqrt(2)*cos(l + pi/4) + sin(2*l)
Подстановка условия
[src]
sin(2*l) + (sin(l) - cos(l))*2 при l = 1/3
подставляем
sin(2*l) + (sin(l) - cos(l))*2
$$\left(\sin{\left(l \right)} - \cos{\left(l \right)}\right) 2 + \sin{\left(2 l \right)}$$
___ / pi\
- 2*\/ 2 *cos|l + --| + sin(2*l)
\ 4 /
$$\sin{\left(2 l \right)} - 2 \sqrt{2} \cos{\left(l + \frac{\pi}{4} \right)}$$
переменные
l = 1/3
$$l = \frac{1}{3}$$
___ / pi\
- 2*\/ 2 *cos|(1/3) + --| + sin(2*(1/3))
\ 4 /
$$\sin{\left(2 (1/3) \right)} - 2 \sqrt{2} \cos{\left((1/3) + \frac{\pi}{4} \right)}$$
___ /1 pi\
- 2*\/ 2 *cos|- + --| + sin(2*1/3)
\3 4 /
$$- 2 \sqrt{2} \cos{\left(\frac{1}{3} + \frac{\pi}{4} \right)} + \sin{\left(2 \cdot \frac{1}{3} \right)}$$
___ /1 pi\
- 2*\/ 2 *cos|- + --| + sin(2/3)
\3 4 /
$$- 2 \sqrt{2} \cos{\left(\frac{1}{3} + \frac{\pi}{4} \right)} + \sin{\left(\frac{2}{3} \right)}$$
-2*sqrt(2)*cos(1/3 + pi/4) + sin(2/3)
Комбинаторика
[src]
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)
Объединение рациональных выражений
[src]
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)
Собрать выражение
[src]
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)
Степени
[src]
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
/ -2*I*l 2*I*l\
I*l -I*l / -I*l I*l\ I*\- e + e /
- e - e - I*\- e + e / - ----------------------
2
$$- i \left(e^{i l} - e^{- i l}\right) - \frac{i \left(e^{2 i l} - e^{- 2 i l}\right)}{2} - e^{i l} - e^{- i l}$$
-exp(i*l) - exp(-i*l) - i*(-exp(-i*l) + exp(i*l)) - i*(-exp(-2*i*l) + exp(2*i*l))/2
Тригонометрическая часть
[src]
___ / pi\
- 2*\/ 2 *cos|l + --| + sin(2*l)
\ 4 /
$$\sin{\left(2 l \right)} - 2 \sqrt{2} \cos{\left(l + \frac{\pi}{4} \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
___ / 3*pi\
- 2*\/ 2 *sin|l + ----| + sin(2*l)
\ 4 /
$$\sin{\left(2 l \right)} - 2 \sqrt{2} \sin{\left(l + \frac{3 \pi}{4} \right)}$$
/ pi\
- 2*sin|l + --| + 2*sin(l) + sin(2*l)
\ 2 /
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \sin{\left(l + \frac{\pi}{2} \right)}$$
___ / pi\ / pi\
- 2*\/ 2 *cos|l + --| + cos|2*l - --|
\ 4 / \ 2 /
$$- 2 \sqrt{2} \cos{\left(l + \frac{\pi}{4} \right)} + \cos{\left(2 l - \frac{\pi}{2} \right)}$$
2/l\
2 - 4*cos |-| + 2*sin(l) + sin(2*l)
\2/
$$- 4 \cos^{2}{\left(\frac{l}{2} \right)} + 2 \sin{\left(l \right)} + \sin{\left(2 l \right)} + 2$$
___
1 2*\/ 2
-------- - ------------
csc(2*l) / pi\
csc|-l + --|
\ 4 /
$$- \frac{2 \sqrt{2}}{\csc{\left(- l + \frac{\pi}{4} \right)}} + \frac{1}{\csc{\left(2 l \right)}}$$
-2*cos(l) + 2*sin(l) + 2*cos(l)*sin(l)
$$2 \sin{\left(l \right)} \cos{\left(l \right)} + 2 \sin{\left(l \right)} - 2 \cos{\left(l \right)}$$
1 2 2 -------- - ------ + ------ csc(2*l) sec(l) csc(l)
$$- \frac{2}{\sec{\left(l \right)}} + \frac{1}{\csc{\left(2 l \right)}} + \frac{2}{\csc{\left(l \right)}}$$
___
1 2*\/ 2
------------- - -----------
/ pi\ / pi\
sec|2*l - --| sec|l + --|
\ 2 / \ 4 /
$$\frac{1}{\sec{\left(2 l - \frac{\pi}{2} \right)}} - \frac{2 \sqrt{2}}{\sec{\left(l + \frac{\pi}{4} \right)}}$$
/ pi\ / pi\
-2*cos(l) + 2*cos|l - --| + cos|2*l - --|
\ 2 / \ 2 /
$$- 2 \cos{\left(l \right)} + 2 \cos{\left(l - \frac{\pi}{2} \right)} + \cos{\left(2 l - \frac{\pi}{2} \right)}$$
1 2 2
-------- - ----------- + ------
csc(2*l) /pi \ csc(l)
csc|-- - l|
\2 /
$$- \frac{2}{\csc{\left(- l + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 l \right)}} + \frac{2}{\csc{\left(l \right)}}$$
/l\
-2*cos(l) + 2*(1 + cos(l))*tan|-| + sin(2*l)
\2/
$$2 \left(\cos{\left(l \right)} + 1\right) \tan{\left(\frac{l}{2} \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
1 2 2 ------------- - ------ + ----------- / pi\ sec(l) / pi\ sec|2*l - --| sec|l - --| \ 2 / \ 2 /
$$\frac{1}{\sec{\left(2 l - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(l - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(l \right)}}$$
1 2 2 ------------- - ------ + ----------- /pi \ sec(l) /pi \ sec|-- - 2*l| sec|-- - l| \2 / \2 /
$$\frac{2}{\sec{\left(- l + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- 2 l + \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(l \right)}}$$
1 2 2
------------- - ----------- + -----------
csc(pi - 2*l) /pi \ csc(pi - l)
csc|-- - l|
\2 /
$$- \frac{2}{\csc{\left(- l + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(- l + \pi \right)}} + \frac{1}{\csc{\left(- 2 l + \pi \right)}}$$
/ 2/l pi\\
-2*cos(l) + |1 - cot |- + --||*(1 + sin(l)) + sin(2*l)
\ \2 4 //
$$\left(- \cot^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(l \right)} + 1\right) + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
___ / 2/l pi\\
2*\/ 2 *|1 - tan |- + --||
\ \2 8 //
- -------------------------- + sin(2*l)
2/l pi\
1 + tan |- + --|
\2 8 /
$$\sin{\left(2 l \right)} - \frac{2 \sqrt{2} \cdot \left(- \tan^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} + 1\right)}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} + 1}$$
___ / 2/l pi\\
2*\/ 2 *|1 - tan |- + --||
2*tan(l) \ \2 8 //
----------- - --------------------------
2 2/l pi\
1 + tan (l) 1 + tan |- + --|
\2 8 /
$$- \frac{2 \sqrt{2} \cdot \left(- \tan^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} + 1\right)}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} + 1} + \frac{2 \tan{\left(l \right)}}{\tan^{2}{\left(l \right)} + 1}$$
/ 2/l\\ /l\
2*|1 - tan |-|| 4*tan|-|
\ \2// 2*tan(l) \2/
- --------------- + ----------- + -----------
2/l\ 2 2/l\
1 + tan |-| 1 + tan (l) 1 + tan |-|
\2/ \2/
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{l}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} + \frac{2 \tan{\left(l \right)}}{\tan^{2}{\left(l \right)} + 1} + \frac{4 \tan{\left(\frac{l}{2} \right)}}{\tan^{2}{\left(\frac{l}{2} \right)} + 1}$$
/l pi\ /l\
4*tan|- + --| 4*cot|-|
\2 4 / 2*cot(l) \2/
- ---------------- + ----------- + -----------
2/l pi\ 2 2/l\
1 + tan |- + --| 1 + cot (l) 1 + cot |-|
\2 4 / \2/
$$\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} + \frac{4 \cot{\left(\frac{l}{2} \right)}}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{l}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1}$$
/l pi\ /l\
4*tan|- + --| 4*tan|-|
\2 4 / 2*tan(l) \2/
- ---------------- + ----------- + -----------
2/l pi\ 2 2/l\
1 + tan |- + --| 1 + tan (l) 1 + tan |-|
\2 4 / \2/
$$- \frac{4 \tan{\left(\frac{l}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(l \right)}}{\tan^{2}{\left(l \right)} + 1} + \frac{4 \tan{\left(\frac{l}{2} \right)}}{\tan^{2}{\left(\frac{l}{2} \right)} + 1}$$
2/l\
8*sin |-|*sin(l)
4*(-1 - cos(2*l) + 2*cos(l)) \2/
- ------------------------------ + ------------------- + sin(2*l)
2 2 4/l\
1 - cos(2*l) + 2*(1 - cos(l)) sin (l) + 4*sin |-|
\2/
$$\frac{8 \sin^{2}{\left(\frac{l}{2} \right)} \sin{\left(l \right)}}{4 \sin^{4}{\left(\frac{l}{2} \right)} + \sin^{2}{\left(l \right)}} + \sin{\left(2 l \right)} - \frac{4 \cdot \left(2 \cos{\left(l \right)} - \cos{\left(2 l \right)} - 1\right)}{2 \left(- \cos{\left(l \right)} + 1\right)^{2} - \cos{\left(2 l \right)} + 1}$$
/ 1 \
2*|1 - -------|
| 2/l\|
| cot |-||
\ \2// 2 4
- --------------- + -------------------- + --------------------
1 / 1 \ / 1 \ /l\
1 + ------- |1 + -------|*cot(l) |1 + -------|*cot|-|
2/l\ | 2 | | 2/l\| \2/
cot |-| \ cot (l)/ | cot |-||
\2/ \ \2//
$$- \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{l}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{l}{2} \right)}}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(l \right)}}\right) \cot{\left(l \right)}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{l}{2} \right)}}\right) \cot{\left(\frac{l}{2} \right)}}$$
2/ pi\ / 2/l\\ / 2/l pi\\
-1 + tan |l + --| 2*|-1 + cot |-|| 2*|-1 + tan |- + --||
\ 4 / \ \2// \ \2 4 //
----------------- - ---------------- + ---------------------
2/ pi\ 2/l\ 2/l pi\
1 + tan |l + --| 1 + cot |-| 1 + tan |- + --|
\ 4 / \2/ \2 4 /
$$\frac{2 \left(\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1} + \frac{\tan^{2}{\left(l + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1} - \frac{2 \left(\cot^{2}{\left(\frac{l}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{l}{2} \right)} + 1}$$
2/ pi\ / 2/l\\ / 2/l pi\\
1 - cot |l + --| 2*|1 - tan |-|| 2*|1 - cot |- + --||
\ 4 / \ \2// \ \2 4 //
---------------- - --------------- + --------------------
2/ pi\ 2/l\ 2/l pi\
1 + cot |l + --| 1 + tan |-| 1 + cot |- + --|
\ 4 / \2/ \2 4 /
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{l}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} + \frac{2 \cdot \left(- \cot^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1} + \frac{- \cot^{2}{\left(l + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(l + \frac{\pi}{4} \right)} + 1}$$
// 1 for l mod 2*pi = 0\ // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
- 2*|< | + 2*|< | + |< |
\\cos(l) otherwise / \\sin(l) otherwise / \\sin(2*l) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\sin{\left(l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\cos{\left(l \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \
|| 1 for |l + --| mod 2*pi = 0|
___ || \ 4 / | // 0 for 2*l mod pi = 0\
- 2*\/ 2 *|< | + |< |
|| 2/l pi\ / 2/l pi\\ | \\sin(2*l) otherwise /
||sin |- + --|*|-1 + cot |- + --|| otherwise |
\\ \2 8 / \ \2 8 // /
$$\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(l + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\
|| | // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
- 2*|< / pi\ | + 2*|< | + |< |
||sin|l + --| otherwise | \\sin(l) otherwise / \\sin(2*l) otherwise /
\\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\sin{\left(l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\sin{\left(l + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
// 1 for l mod 2*pi = 0\ || | || |
- 2*|< | + 2*|< / pi\ | + |< / pi\ |
\\cos(l) otherwise / ||cos|l - --| otherwise | ||cos|2*l - --| otherwise |
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(l - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\cos{\left(2 l - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\cos{\left(l \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \
|| 1 for |l + --| mod 2*pi = 0|
|| \ 4 / | // 0 for 2*l mod pi = 0\
|| | || |
___ || 2/l pi\ | || 2*cot(l) |
- 2*\/ 2 *|<-1 + cot |- + --| | + |<----------- otherwise |
|| \2 8 / | || 2 |
||----------------- otherwise | ||1 + cot (l) |
|| 2/l pi\ | \\ /
|| 1 + cot |- + --| |
\\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(2 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(l + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for l mod pi = 0\
|| |
// 1 for l mod 2*pi = 0\ ||1 - cos(l) | // 0 for 2*l mod pi = 0\
- 2*|< | + 2*|<---------- otherwise | + |< |
\\cos(l) otherwise / || /l\ | \\sin(2*l) otherwise /
|| tan|-| |
\\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{- \cos{\left(l \right)} + 1}{\tan{\left(\frac{l}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\cos{\left(l \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\
|| | // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
|| 1 | || | || |
- 2*|<----------- otherwise | + 2*|< 1 | + |< 1 |
|| /pi \ | ||------ otherwise | ||-------- otherwise |
||csc|-- - l| | \\csc(l) / \\csc(2*l) /
\\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{1}{\csc{\left(l \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{1}{\csc{\left(2 l \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- l + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
// 1 for l mod 2*pi = 0\ || | || |
|| | || 1 | || 1 |
- 2*|< 1 | + 2*|<----------- otherwise | + |<------------- otherwise |
||------ otherwise | || / pi\ | || / pi\ |
\\sec(l) / ||sec|l - --| | ||sec|2*l - --| |
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{1}{\sec{\left(l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{1}{\sec{\left(2 l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(l \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for l mod 2*pi = 0\ || 1 for |l + ----| mod 2*pi = 0| || 1 for |2*l + ----| mod 2*pi = 0|
- 2*|< | + 2*|< \ 2 / | + |< \ 2 / |
\\cos(l) otherwise / || | || |
\\sin(l) otherwise / \\sin(2*l) otherwise /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\cos{\left(l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
|| 0 for |l + --| mod pi = 0|
|| \ 2 / | // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
- 2*|< | + 2*|< | + |< |
|| /l pi\ | \\sin(l) otherwise / \\sin(2*l) otherwise /
||(1 + sin(l))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\sin{\left(l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: \left(l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(l \right)} + 1\right) \cot{\left(\frac{l}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/l\\
| 4*sin |-||
| \2/|
2*|1 - ---------| 2/l\
| 2 | 2 8*sin |-|
\ sin (l) / 4*sin (l) \2/
- ----------------- + ------------------------ + ----------------------
4/l\ / 4 \ / 4/l\\
4*sin |-| | 4*sin (l)| | 4*sin |-||
\2/ |1 + ---------|*sin(2*l) | \2/|
1 + --------- | 2 | |1 + ---------|*sin(l)
2 \ sin (2*l)/ | 2 |
sin (l) \ sin (l) /
$$- \frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{l}{2} \right)}}{\sin^{2}{\left(l \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{l}{2} \right)}}{\sin^{2}{\left(l \right)}} + 1} + \frac{4 \sin^{2}{\left(l \right)}}{\left(\frac{4 \sin^{4}{\left(l \right)}}{\sin^{2}{\left(2 l \right)}} + 1\right) \sin{\left(2 l \right)}} + \frac{8 \sin^{2}{\left(\frac{l}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{l}{2} \right)}}{\sin^{2}{\left(l \right)}} + 1\right) \sin{\left(l \right)}}$$
// 1 for l mod 2*pi = 0\ // 0 for l mod pi = 0\
|| | || | // 0 for 2*l mod pi = 0\
|| 2/l\ | || /l\ | || |
||-1 + cot |-| | || 2*cot|-| | || 2*cot(l) |
- 2*|< \2/ | + 2*|< \2/ | + |<----------- otherwise |
||------------ otherwise | ||----------- otherwise | || 2 |
|| 2/l\ | || 2/l\ | ||1 + cot (l) |
||1 + cot |-| | ||1 + cot |-| | \\ /
\\ \2/ / \\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{l}{2} \right)}}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{l}{2} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\ // 0 for l mod pi = 0\
|| | || | // 0 for 2*l mod pi = 0\
|| 2/l\ | || /l\ | || |
||1 - tan |-| | || 2*tan|-| | || 2*tan(l) |
- 2*|< \2/ | + 2*|< \2/ | + |<----------- otherwise |
||----------- otherwise | ||----------- otherwise | || 2 |
|| 2/l\ | || 2/l\ | ||1 + tan (l) |
||1 + tan |-| | ||1 + tan |-| | \\ /
\\ \2/ / \\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{l}{2} \right)}}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \tan{\left(l \right)}}{\tan^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{l}{2} \right)} + 1}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/l\ \
| sec |-| |
| \2/ |
2*|1 - ------------|
| 2/l pi\| /l\
| sec |- - --|| 4*sec|-|
\ \2 2 // 2*sec(l) \2/
- -------------------- + ------------------------------ + ------------------------------
2/l\ / 2 \ / 2/l\ \
sec |-| | sec (l) | / pi\ | sec |-| |
\2/ |1 + ------------|*sec|l - --| | \2/ | /l pi\
1 + ------------ | 2/ pi\| \ 2 / |1 + ------------|*sec|- - --|
2/l pi\ | sec |l - --|| | 2/l pi\| \2 2 /
sec |- - --| \ \ 2 // | sec |- - --||
\2 2 / \ \2 2 //
$$- \frac{2 \left(- \frac{\sec^{2}{\left(\frac{l}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{l}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{2 \sec{\left(l \right)}}{\left(\frac{\sec^{2}{\left(l \right)}}{\sec^{2}{\left(l - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(l - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(\frac{l}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{l}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}$$
/ 2/l pi\\
| cos |- - --||
| \2 2 /|
2*|1 - ------------|
| 2/l\ | / pi\ /l pi\
| cos |-| | 2*cos|l - --| 4*cos|- - --|
\ \2/ / \ 2 / \2 2 /
- -------------------- + ------------------------- + -------------------------
2/l pi\ / 2/ pi\\ / 2/l pi\\
cos |- - --| | cos |l - --|| | cos |- - --||
\2 2 / | \ 2 /| | \2 2 /| /l\
1 + ------------ |1 + ------------|*cos(l) |1 + ------------|*cos|-|
2/l\ | 2 | | 2/l\ | \2/
cos |-| \ cos (l) / | cos |-| |
\2/ \ \2/ /
$$- \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} \right)}}} + \frac{2 \cos{\left(l - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(l \right)}}\right) \cos{\left(l \right)}} + \frac{4 \cos{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} \right)}}\right) \cos{\left(\frac{l}{2} \right)}}$$
// / pi\ \
|| 0 for |l + --| mod pi = 0| // 0 for l mod pi = 0\
|| \ 2 / | || | // 0 for 2*l mod pi = 0\
|| | || /l\ | || |
|| /l pi\ | || 2*cot|-| | || 2*cot(l) |
- 2*|< 2*cot|- + --| | + 2*|< \2/ | + |<----------- otherwise |
|| \2 4 / | ||----------- otherwise | || 2 |
||---------------- otherwise | || 2/l\ | ||1 + cot (l) |
|| 2/l pi\ | ||1 + cot |-| | \\ /
||1 + cot |- + --| | \\ \2/ /
\\ \2 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{l}{2} \right)}}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: \left(l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{l}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi l\\
| csc |-- - -||
| \2 2/|
2*|1 - ------------|
| 2/l\ | /pi \ /pi l\
| csc |-| | 2*csc|-- - l| 4*csc|-- - -|
\ \2/ / \2 / \2 2/
- -------------------- + ------------------------- + -------------------------
2/pi l\ / 2/pi \\ / 2/pi l\\
csc |-- - -| | csc |-- - l|| | csc |-- - -||
\2 2/ | \2 /| | \2 2/| /l\
1 + ------------ |1 + ------------|*csc(l) |1 + ------------|*csc|-|
2/l\ | 2 | | 2/l\ | \2/
csc |-| \ csc (l) / | csc |-| |
\2/ \ \2/ /
$$- \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{l}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{l}{2} \right)}}} + \frac{2 \csc{\left(- l + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(l \right)}}\right) \csc{\left(l \right)}} + \frac{4 \csc{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{l}{2} \right)}}\right) \csc{\left(\frac{l}{2} \right)}}$$
// 1 for l mod 2*pi = 0\
|| |
|| 1 | // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
||-1 + ------- | || | || |
|| 2/l\ | || 2 | || 2 |
|| tan |-| | ||-------------------- otherwise | ||-------------------- otherwise |
- 2*|< \2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{l}{2} \right)}}\right) \tan{\left(\frac{l}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(l \right)}}\right) \tan{\left(l \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{l}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{l}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\ // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
|| | || | || |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\sin{\left(l \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\cos{\left(l \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for l mod pi = 0\
|| |
// 1 for l mod 2*pi = 0\ || 2*sin(l) |
|| | ||---------------------------- otherwise |
|| 2 | || / 2 \ | // 0 for 2*l mod pi = 0\
- 2*|< -4 + 4*sin (l) + 4*cos(l) | + 2*|< | sin (l) | | + |< |
||--------------------------- otherwise | ||(1 - cos(l))*|1 + ---------| | \\sin(2*l) otherwise /
|| 2 2 | || | 4/l\| |
\\2*(1 - cos(l)) + 2*sin (l) / || | 4*sin |-|| |
|| \ \2// |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \sin{\left(l \right)}}{\left(1 + \frac{\sin^{2}{\left(l \right)}}{4 \sin^{4}{\left(\frac{l}{2} \right)}}\right) \left(- \cos{\left(l \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(l \right)} + 4 \cos{\left(l \right)} - 4}{2 \left(- \cos{\left(l \right)} + 1\right)^{2} + 2 \sin^{2}{\left(l \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for l mod 2*pi = 0\ || 1 for |l + ----| mod 2*pi = 0| || 1 for |2*l + ----| mod 2*pi = 0|
|| | || \ 2 / | || \ 2 / |
|| 2/l\ | || | || |
||-1 + cot |-| | || 2/l pi\ | || 2/ pi\ |
- 2*|< \2/ | + 2*|<-1 + tan |- + --| | + |<-1 + tan |l + --| |
||------------ otherwise | || \2 4 / | || \ 4 / |
|| 2/l\ | ||----------------- otherwise | ||----------------- otherwise |
||1 + cot |-| | || 2/l pi\ | || 2/ pi\ |
\\ \2/ / || 1 + tan |- + --| | || 1 + tan |l + --| |
\\ \2 4 / / \\ \ 4 / /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{l}{2} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{l}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(l + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for l mod 2*pi = 0\
|| |
|| 2 | // 0 for l mod pi = 0\
|| sin (l) | || | // 0 for 2*l mod pi = 0\
||-1 + --------- | || sin(l) | || |
|| 4/l\ | ||----------------------- otherwise | || sin(2*l) |
|| 4*sin |-| | ||/ 2 \ | ||----------------------- otherwise |
- 2*|< \2/ | + 2*|<| sin (l) | 2/l\ | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{\sin{\left(l \right)}}{\left(1 + \frac{\sin^{2}{\left(l \right)}}{4 \sin^{4}{\left(\frac{l}{2} \right)}}\right) \sin^{2}{\left(\frac{l}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\sin{\left(2 l \right)}}{\left(1 + \frac{\sin^{2}{\left(2 l \right)}}{4 \sin^{4}{\left(l \right)}}\right) \sin^{2}{\left(l \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(l \right)}}{4 \sin^{4}{\left(\frac{l}{2} \right)}}}{1 + \frac{\sin^{2}{\left(l \right)}}{4 \sin^{4}{\left(\frac{l}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\ // 0 for l mod pi = 0\
|| | || | // 0 for 2*l mod pi = 0\
||/ 1 for l mod 2*pi = 0 | ||/ 0 for l mod pi = 0 | || |
||| | ||| | ||/ 0 for 2*l mod pi = 0 |
||| 2/l\ | ||| /l\ | ||| |
- 2*|<|-1 + cot |-| | + 2*|<| 2*cot|-| | + |<| 2*cot(l) |
||< \2/ otherwise | ||< \2/ otherwise | ||<----------- otherwise otherwise |
|||------------ otherwise | |||----------- otherwise | ||| 2 |
||| 2/l\ | ||| 2/l\ | |||1 + cot (l) |
|||1 + cot |-| | |||1 + cot |-| | \\\ /
\\\ \2/ / \\\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{l}{2} \right)}}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{l}{2} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\
|| |
|| 2/l\ | // 0 for l mod pi = 0\
|| cos |-| | || | // 0 for 2*l mod pi = 0\
|| \2/ | || /l\ | || |
||-1 + ------------ | || 2*cos|-| | || 2*cos(l) |
|| 2/l pi\ | || \2/ | ||------------------------------ otherwise |
|| cos |- - --| | ||------------------------------ otherwise | ||/ 2 \ |
- 2*|< \2 2 / | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{l}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{l}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cos{\left(l \right)}}{\left(\frac{\cos^{2}{\left(l \right)}}{\cos^{2}{\left(l - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{l}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{l}{2} \right)}}{\cos^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\
|| |
|| 2/l pi\ | // 0 for l mod pi = 0\ // 0 for 2*l mod pi = 0\
|| sec |- - --| | || | || |
|| \2 2 / | || /l pi\ | || / pi\ |
||-1 + ------------ | || 2*sec|- - --| | || 2*sec|l - --| |
|| 2/l\ | || \2 2 / | || \ 2 / |
|| sec |-| | ||------------------------- otherwise | ||------------------------- otherwise |
- 2*|< \2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} \right)}}\right) \sec{\left(\frac{l}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \sec{\left(l - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(l \right)}}\right) \sec{\left(l \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{l}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{l}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for l mod 2*pi = 0\
|| |
|| 2/l\ | // 0 for l mod pi = 0\
|| csc |-| | || | // 0 for 2*l mod pi = 0\
|| \2/ | || /l\ | || |
||-1 + ------------ | || 2*csc|-| | || 2*csc(l) |
|| 2/pi l\ | || \2/ | ||------------------------------ otherwise |
|| csc |-- - -| | ||------------------------------ otherwise | ||/ 2 \ |
- 2*|< \2 2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: l \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{l}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{l}{2} \right)}}{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \csc{\left(l \right)}}{\left(\frac{\csc^{2}{\left(l \right)}}{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- l + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: l \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{l}{2} \right)}}{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{l}{2} \right)}}{\csc^{2}{\left(- \frac{l}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
-2*Piecewise((1, Mod(l = 2*pi, 0)), ((-1 + csc(l/2)^2/csc(pi/2 - l/2)^2)/(1 + csc(l/2)^2/csc(pi/2 - l/2)^2), True)) + 2*Piecewise((0, Mod(l = pi, 0)), (2*csc(l/2)/((1 + csc(l/2)^2/csc(pi/2 - l/2)^2)*csc(pi/2 - l/2)), True)) + Piecewise((0, Mod(2*l = pi, 0)), (2*csc(l)/((1 + csc(l)^2/csc(pi/2 - l)^2)*csc(pi/2 - l)), True))
Общий знаменатель
[src]
-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)
Рациональный знаменатель
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-2*cos(l) + 2*sin(l) + sin(2*l)
$$2 \sin{\left(l \right)} + \sin{\left(2 l \right)} - 2 \cos{\left(l \right)}$$
-2*cos(l) + 2*sin(l) + sin(2*l)