Господин Экзамен
cos(4*l)+sin(2*l)*cos(2*l) если l=-1/4
Решение
Вы ввели
[src]
cos(4*l) + sin(2*l)*cos(2*l)
$$\sin{\left(2 l \right)} \cos{\left(2 l \right)} + \cos{\left(4 l \right)}$$
cos(4*l) + sin(2*l)*cos(2*l)
Общее упрощение
[src]
sin(4*l) -------- + cos(4*l) 2
$$\frac{\sin{\left(4 l \right)}}{2} + \cos{\left(4 l \right)}$$
sin(4*l)/2 + cos(4*l)
Подстановка условия
[src]
cos(4*l) + sin(2*l)*cos(2*l) при l = -1/4
подставляем
cos(4*l) + sin(2*l)*cos(2*l)
$$\sin{\left(2 l \right)} \cos{\left(2 l \right)} + \cos{\left(4 l \right)}$$
sin(4*l) -------- + cos(4*l) 2
$$\frac{\sin{\left(4 l \right)}}{2} + \cos{\left(4 l \right)}$$
переменные
l = -1/4
$$l = - \frac{1}{4}$$
sin(4*(-1/4))
------------- + cos(4*(-1/4))
2
$$\frac{\sin{\left(4 (-1/4) \right)}}{2} + \cos{\left(4 (-1/4) \right)}$$
sin(1)
- ------ + cos(1)
2
$$- \frac{\sin{\left(1 \right)}}{2} + \cos{\left(1 \right)}$$
-sin(1)/2 + cos(1)
Собрать выражение
[src]
sin(4*l) -------- + cos(4*l) 2
$$\frac{\sin{\left(4 l \right)}}{2} + \cos{\left(4 l \right)}$$
sin(4*l)/2 + cos(4*l)
Степени
[src]
/ -2*I*l 2*I*l\
|e e | / -2*I*l 2*I*l\
-4*I*l 4*I*l I*|------- + ------|*\- e + e /
e e \ 2 2 /
------- + ------ - -----------------------------------------
2 2 2
$$\frac{e^{4 i l}}{2} - \frac{i \left(\frac{e^{2 i l}}{2} + \frac{e^{- 2 i l}}{2}\right) \left(e^{2 i l} - e^{- 2 i l}\right)}{2} + \frac{e^{- 4 i l}}{2}$$
exp(-4*i*l)/2 + exp(4*i*l)/2 - i*(exp(-2*i*l)/2 + exp(2*i*l)/2)*(-exp(-2*i*l) + exp(2*i*l))/2
Тригонометрическая часть
[src]
sin(4*l) -------- + cos(4*l) 2
$$\frac{\sin{\left(4 l \right)}}{2} + \cos{\left(4 l \right)}$$
1 sin(4*l) -------- + -------- sec(4*l) 2
$$\frac{\sin{\left(4 l \right)}}{2} + \frac{1}{\sec{\left(4 l \right)}}$$
1 1 -------- + ---------- sec(4*l) 2*csc(4*l)
$$\frac{1}{\sec{\left(4 l \right)}} + \frac{1}{2 \csc{\left(4 l \right)}}$$
sin(4*l) /pi \ -------- + sin|-- + 4*l| 2 \2 /
$$\frac{\sin{\left(4 l \right)}}{2} + \sin{\left(4 l + \frac{\pi}{2} \right)}$$
/ pi\
cos|4*l - --|
\ 2 /
------------- + cos(4*l)
2
$$\cos{\left(4 l \right)} + \frac{\cos{\left(4 l - \frac{\pi}{2} \right)}}{2}$$
1 1
-------- + ---------------
sec(4*l) / pi\
2*sec|4*l - --|
\ 2 /
$$\frac{1}{2 \sec{\left(4 l - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 l \right)}}$$
1 sin(4*l) ------------- + -------- /pi \ 2 csc|-- - 4*l| \2 /
$$\frac{\sin{\left(4 l \right)}}{2} + \frac{1}{\csc{\left(- 4 l + \frac{\pi}{2} \right)}}$$
/ pi\
cos(2*l)*cos|2*l - --| + cos(4*l)
\ 2 /
$$\cos{\left(2 l \right)} \cos{\left(2 l - \frac{\pi}{2} \right)} + \cos{\left(4 l \right)}$$
1 1 ------------- + ---------- /pi \ 2*csc(4*l) csc|-- - 4*l| \2 /
$$\frac{1}{\csc{\left(- 4 l + \frac{\pi}{2} \right)}} + \frac{1}{2 \csc{\left(4 l \right)}}$$
1 1 -------- + ----------------- sec(4*l) csc(2*l)*sec(2*l)
$$\frac{1}{\sec{\left(4 l \right)}} + \frac{1}{\csc{\left(2 l \right)} \sec{\left(2 l \right)}}$$
/pi \ /pi \
sin(2*l)*sin|-- + 2*l| + sin|-- + 4*l|
\2 / \2 /
$$\sin{\left(2 l \right)} \sin{\left(2 l + \frac{\pi}{2} \right)} + \sin{\left(4 l + \frac{\pi}{2} \right)}$$
2
sin(4*l) 1 - tan (2*l)
-------- + -------------
2 2
sec (2*l)
$$\frac{\sin{\left(4 l \right)}}{2} + \frac{- \tan^{2}{\left(2 l \right)} + 1}{\sec^{2}{\left(2 l \right)}}$$
2
-1 + 2*cos (2*l) + cos(4*l) sin(4*l)
--------------------------- + --------
2 2
$$\frac{2 \cos^{2}{\left(2 l \right)} + \cos{\left(4 l \right)} - 1}{2} + \frac{\sin{\left(4 l \right)}}{2}$$
1 1
-------- + ----------------------
sec(4*l) / pi\
sec(2*l)*sec|2*l - --|
\ 2 /
$$\frac{1}{\sec{\left(4 l \right)}} + \frac{1}{\sec{\left(2 l \right)} \sec{\left(2 l - \frac{\pi}{2} \right)}}$$
sin(4*l) 2 / 2 \ -------- + cos (2*l)*\1 - tan (2*l)/ 2
$$\left(- \tan^{2}{\left(2 l \right)} + 1\right) \cos^{2}{\left(2 l \right)} + \frac{\sin{\left(4 l \right)}}{2}$$
1 1
-------- + ----------------------
sec(4*l) /pi \
sec(2*l)*sec|-- - 2*l|
\2 /
$$\frac{1}{\sec{\left(4 l \right)}} + \frac{1}{\sec{\left(2 l \right)} \sec{\left(- 2 l + \frac{\pi}{2} \right)}}$$
sin(4*l) 2
-3 + -------- + 2*(1 - cos(2*l)) + 4*cos(2*l)
2
$$2 \left(- \cos{\left(2 l \right)} + 1\right)^{2} + \frac{\sin{\left(4 l \right)}}{2} + 4 \cos{\left(2 l \right)} - 3$$
1 1 ------------- + ---------------------- /pi \ /pi \ csc|-- - 4*l| csc(2*l)*csc|-- - 2*l| \2 / \2 /
$$\frac{1}{\csc{\left(- 4 l + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 l \right)} \csc{\left(- 2 l + \frac{\pi}{2} \right)}}$$
1 1 ------------- + --------------------------- /pi \ /pi \ csc|-- - 4*l| csc(pi - 2*l)*csc|-- - 2*l| \2 / \2 /
$$\frac{1}{\csc{\left(- 4 l + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 l + \pi \right)} \csc{\left(- 2 l + \frac{\pi}{2} \right)}}$$
2
1 - tan (2*l) tan(2*l)
------------- + -------------
2 2
1 + tan (2*l) 1 + tan (2*l)
$$\frac{- \tan^{2}{\left(2 l \right)} + 1}{\tan^{2}{\left(2 l \right)} + 1} + \frac{\tan{\left(2 l \right)}}{\tan^{2}{\left(2 l \right)} + 1}$$
/ 0 for 4*l mod pi = 0
<
\sin(4*l) otherwise
----------------------------- + cos(4*l)
2
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\sin{\left(4 l \right)} & \text{otherwise} \end{cases}}{2}\right) + \cos{\left(4 l \right)}$$
2 sin(4*l) / 2 \ 4 / 2 \ -------- + \1 - tan (l)/ *cos (l)*\1 - tan (2*l)/ 2
$$\left(- \tan^{2}{\left(l \right)} + 1\right)^{2} \cdot \left(- \tan^{2}{\left(2 l \right)} + 1\right) \cos^{4}{\left(l \right)} + \frac{\sin{\left(4 l \right)}}{2}$$
/ pi\
2*tan|2*l + --|
tan(2*l) \ 4 /
------------- + ------------------
2 2/ pi\
1 + tan (2*l) 1 + tan |2*l + --|
\ 4 /
$$\frac{2 \tan{\left(2 l + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1} + \frac{\tan{\left(2 l \right)}}{\tan^{2}{\left(2 l \right)} + 1}$$
/ 4 \
sin(4*l) 2/pi \ | 4*sin (2*l)|
-------- + sin |-- + 2*l|*|1 - -----------|
2 \2 / | 2 |
\ sin (4*l) /
$$\left(- \frac{4 \sin^{4}{\left(2 l \right)}}{\sin^{2}{\left(4 l \right)}} + 1\right) \sin^{2}{\left(2 l + \frac{\pi}{2} \right)} + \frac{\sin{\left(4 l \right)}}{2}$$
2
sec (2*l)
1 - --------------
2/ pi\
sec |2*l - --|
1 \ 2 /
--------------- + ------------------
/ pi\ 2
2*sec|4*l - --| sec (2*l)
\ 2 /
$$\frac{- \frac{\sec^{2}{\left(2 l \right)}}{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}} + 1}{\sec^{2}{\left(2 l \right)}} + \frac{1}{2 \sec{\left(4 l - \frac{\pi}{2} \right)}}$$
/ pi\ / 2/ pi\\
cos|4*l - --| | cos |2*l - --||
\ 2 / 2 | \ 2 /|
------------- + cos (2*l)*|1 - --------------|
2 | 2 |
\ cos (2*l) /
$$\left(1 - \frac{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 l \right)}}\right) \cos^{2}{\left(2 l \right)} + \frac{\cos{\left(4 l - \frac{\pi}{2} \right)}}{2}$$
2/pi \
csc |-- - 2*l|
\2 /
1 - --------------
2
1 csc (2*l)
---------- + ------------------
2*csc(4*l) 2/pi \
csc |-- - 2*l|
\2 /
$$\frac{1 - \frac{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 l \right)}}}{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}} + \frac{1}{2 \csc{\left(4 l \right)}}$$
2 / 2 \
1 - tan (2*l) 2*\1 - tan (l)/*tan(l)
------------- + ----------------------
2 2
1 + tan (2*l) / 2 \
\1 + tan (l)/
$$\frac{2 \cdot \left(- \tan^{2}{\left(l \right)} + 1\right) \tan{\left(l \right)}}{\left(\tan^{2}{\left(l \right)} + 1\right)^{2}} + \frac{- \tan^{2}{\left(2 l \right)} + 1}{\tan^{2}{\left(2 l \right)} + 1}$$
2
/ 2 \ / 2 \
tan(2*l) \1 - tan (l)/ *\1 - tan (2*l)/
------------- + ------------------------------
2 2
1 + tan (2*l) / 2 \
\1 + tan (l)/
$$\frac{\left(- \tan^{2}{\left(l \right)} + 1\right)^{2} \cdot \left(- \tan^{2}{\left(2 l \right)} + 1\right)}{\left(\tan^{2}{\left(l \right)} + 1\right)^{2}} + \frac{\tan{\left(2 l \right)}}{\tan^{2}{\left(2 l \right)} + 1}$$
1 / 1 \
1 - --------- 2*|1 - -------|
2 | 2 |
cot (2*l) \ cot (l)/
------------- + ---------------------
1 2
1 + --------- / 1 \
2 |1 + -------| *cot(l)
cot (2*l) | 2 |
\ cot (l)/
$$\frac{1 - \frac{1}{\cot^{2}{\left(2 l \right)}}}{1 + \frac{1}{\cot^{2}{\left(2 l \right)}}} + \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(l \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(l \right)}}\right)^{2} \cot{\left(l \right)}}$$
/ 0 for 4*l mod pi = 0
|
| 2*cot(2*l)
<------------- otherwise
| 2
|1 + cot (2*l) 2
\ -1 + cot (2*l)
---------------------------------- + --------------
2 2
1 + cot (2*l)
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\frac{2 \cot{\left(2 l \right)}}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{\cot^{2}{\left(2 l \right)} - 1}{\cot^{2}{\left(2 l \right)} + 1}$$
2 / 2 2 \ 2 2 / 2 2 \ \cos (l) - sin (l)/ - 4*cos (l)*sin (l) + 2*\cos (l) - sin (l)/*cos(l)*sin(l)
$$- 4 \sin^{2}{\left(l \right)} \cos^{2}{\left(l \right)} + 2 \left(- \sin^{2}{\left(l \right)} + \cos^{2}{\left(l \right)}\right) \sin{\left(l \right)} \cos{\left(l \right)} + \left(- \sin^{2}{\left(l \right)} + \cos^{2}{\left(l \right)}\right)^{2}$$
/ 0 for 4*l mod pi = 0
|
| 2*cot(2*l)
<------------- otherwise
| 2 / pi\
|1 + cot (2*l) 2*tan|2*l + --|
\ \ 4 /
---------------------------------- + ------------------
2 2/ pi\
1 + tan |2*l + --|
\ 4 /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\frac{2 \cot{\left(2 l \right)}}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{2 \tan{\left(2 l + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1}$$
/ pi\ / pi\
2*tan|2*l + --| 4*cot(l)*tan|l + --|
\ 4 / \ 4 /
------------------ + --------------------------------
2/ pi\ / 2 \ / 2/ pi\\
1 + tan |2*l + --| \1 + cot (l)/*|1 + tan |l + --||
\ 4 / \ \ 4 //
$$\frac{2 \tan{\left(2 l + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(l + \frac{\pi}{4} \right)} \cot{\left(l \right)}}{\left(\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(l \right)} + 1\right)}$$
/ pi\ / pi\
2*tan|2*l + --| 4*tan(l)*tan|l + --|
\ 4 / \ 4 /
------------------ + --------------------------------
2/ pi\ / 2 \ / 2/ pi\\
1 + tan |2*l + --| \1 + tan (l)/*|1 + tan |l + --||
\ 4 / \ \ 4 //
$$\frac{2 \tan{\left(2 l + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(l \right)} \tan{\left(l + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(l \right)} + 1\right) \left(\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1\right)}$$
/ 0 for 4*l mod pi = 0
< // /pi \ \
\sin(4*l) otherwise || 0 for |-- + 4*l| mod pi = 0|
----------------------------- + |< \2 / |
2 || |
\\cos(4*l) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\sin{\left(4 l \right)} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 0 & \text{for}\: \left(4 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 l \right)} & \text{otherwise} \end{cases}\right)$$
/ 2 \ / 2/ pi\\
2 \-1 + cot (l)/*|-1 + tan |l + --||
-1 + cot (2*l) \ \ 4 //
-------------- + ----------------------------------
2 / 2 \ / 2/ pi\\
1 + cot (2*l) \1 + cot (l)/*|1 + tan |l + --||
\ \ 4 //
$$\frac{\left(\tan^{2}{\left(l + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(l \right)} - 1\right)}{\left(\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(l \right)} + 1\right)} + \frac{\cot^{2}{\left(2 l \right)} - 1}{\cot^{2}{\left(2 l \right)} + 1}$$
/ 2/ pi\\ / 2 \
2 |1 - cot |l + --||*\1 - tan (l)/
1 - tan (2*l) \ \ 4 //
------------- + --------------------------------
2 / 2/ pi\\ / 2 \
1 + tan (2*l) |1 + cot |l + --||*\1 + tan (l)/
\ \ 4 //
$$\frac{\left(- \tan^{2}{\left(l \right)} + 1\right) \left(- \cot^{2}{\left(l + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(l \right)} + 1\right) \left(\cot^{2}{\left(l + \frac{\pi}{4} \right)} + 1\right)} + \frac{- \tan^{2}{\left(2 l \right)} + 1}{\tan^{2}{\left(2 l \right)} + 1}$$
/ 0 for 4*l mod pi = 0
< // 1 for 2*l mod pi = 0\
\sin(4*l) otherwise || |
----------------------------- + |< 2 / 2 \ |
2 ||sin (2*l)*\-1 + cot (2*l)/ otherwise |
\\ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\sin{\left(4 l \right)} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\cot^{2}{\left(2 l \right)} - 1\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
/ 0 for 4*l mod pi = 0
< // 1 for l mod pi = 0\
\sin(4*l) otherwise / 2 \ || |
----------------------------- + \1 - tan (2*l)/*|<1 + cos(4*l) |
2 ||------------ otherwise |
\\ 2 /
$$\left(\left(- \tan^{2}{\left(2 l \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\cos{\left(4 l \right)} + 1}{2} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\sin{\left(4 l \right)} & \text{otherwise} \end{cases}}{2}\right)$$
/ 0 for 4*l mod pi = 0
|
| 2*cot(2*l)
<------------- otherwise
| 2 // 1 for 2*l mod pi = 0\
|1 + cot (2*l) || |
\ || 2 |
---------------------------------- + |<-1 + cot (2*l) |
2 ||-------------- otherwise |
|| 2 |
\\1 + cot (2*l) /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\frac{2 \cot{\left(2 l \right)}}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 l \right)} - 1}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}\right)$$
4 / 4 \
4*sin (2*l) 2 | 4*sin (l)|
1 - ----------- 4*sin (l)*|1 - ---------|
2 | 2 |
sin (4*l) \ sin (2*l)/
--------------- + -------------------------
4 2
4*sin (2*l) / 4 \
1 + ----------- | 4*sin (l)|
2 |1 + ---------| *sin(2*l)
sin (4*l) | 2 |
\ sin (2*l)/
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(l \right)}}{\sin^{2}{\left(2 l \right)}} + 1\right) \sin^{2}{\left(l \right)}}{\left(\frac{4 \sin^{4}{\left(l \right)}}{\sin^{2}{\left(2 l \right)}} + 1\right)^{2} \sin{\left(2 l \right)}} + \frac{- \frac{4 \sin^{4}{\left(2 l \right)}}{\sin^{2}{\left(4 l \right)}} + 1}{\frac{4 \sin^{4}{\left(2 l \right)}}{\sin^{2}{\left(4 l \right)}} + 1}$$
/ 0 for 4*l mod pi = 0
|
| 2*cot(2*l) // /pi \ \
<------------- otherwise || 0 for |-- + 4*l| mod pi = 0|
| 2 || \2 / |
|1 + cot (2*l) || |
\ || / pi\ |
---------------------------------- + |< 2*cot|2*l + --| |
2 || \ 4 / |
||------------------ otherwise |
|| 2/ pi\ |
||1 + cot |2*l + --| |
\\ \ 4 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\frac{2 \cot{\left(2 l \right)}}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 0 & \text{for}\: \left(4 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 l + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || |
|< |*|< | + |< 2 / 2 \ |
\\sin(2*l) otherwise / \\cos(2*l) otherwise / ||sin (2*l)*\-1 + cot (2*l)/ otherwise |
\\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\cot^{2}{\left(2 l \right)} - 1\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
|| 1 |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ ||-1 + --------- |
|< |*|< | + |< 2 |
\\sin(2*l) otherwise / \\cos(2*l) otherwise / || tan (2*l) |
||-------------- otherwise |
|| 2 |
\\ csc (2*l) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 l \right)}}}{\csc^{2}{\left(2 l \right)}} & \text{otherwise} \end{cases}\right)$$
// /pi \ \ // /pi \ \
// 0 for 2*l mod pi = 0\ || 0 for |-- + 2*l| mod pi = 0| || 0 for |-- + 4*l| mod pi = 0|
|< |*|< \2 / | + |< \2 / |
\\sin(2*l) otherwise / || | || |
\\cos(2*l) otherwise / \\cos(4*l) otherwise /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(4 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 l \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || 2 / 1 \ |
|< |*|< | + |
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 l \right)}}\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
/ 0 for 4*l mod pi = 0
|
| 2*cot(2*l)
<------------- otherwise // 1 for l mod pi = 0\
| 2 || |
|1 + cot (2*l) || 2 |
\ / 1 \ ||/ 2 \ |
---------------------------------- + |1 - ---------|*|<\-1 + cot (l)/ |
2 | 2 | ||--------------- otherwise |
\ cot (2*l)/ || 2 |
|| / 2 \ |
\\ \1 + cot (l)/ /
$$\left(\left(1 - \frac{1}{\cot^{2}{\left(2 l \right)}}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(l \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(l \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{\begin{cases} 0 & \text{for}\: 4 l \bmod \pi = 0 \\\frac{2 \cot{\left(2 l \right)}}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}}{2}\right)$$
2 / 2 \
sec (2*l) | sec (l) |
1 - -------------- 2*|1 - ------------|*sec(l)
2/ pi\ | 2/ pi\|
sec |2*l - --| | sec |l - --||
\ 2 / \ \ 2 //
------------------ + -------------------------------
2 2
sec (2*l) / 2 \
1 + -------------- | sec (l) | / pi\
2/ pi\ |1 + ------------| *sec|l - --|
sec |2*l - --| | 2/ pi\| \ 2 /
\ 2 / | sec |l - --||
\ \ 2 //
$$\frac{- \frac{\sec^{2}{\left(2 l \right)}}{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(2 l \right)}}{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}} + 1} + \frac{2 \left(- \frac{\sec^{2}{\left(l \right)}}{\sec^{2}{\left(l - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(l \right)}}{\left(\frac{\sec^{2}{\left(l \right)}}{\sec^{2}{\left(l - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(l - \frac{\pi}{2} \right)}}$$
2/ pi\ / 2/ pi\\
cos |2*l - --| | cos |l - --||
\ 2 / | \ 2 /| / pi\
1 - -------------- 2*|1 - ------------|*cos|l - --|
2 | 2 | \ 2 /
cos (2*l) \ cos (l) /
------------------ + --------------------------------
2/ pi\ 2
cos |2*l - --| / 2/ pi\\
\ 2 / | cos |l - --||
1 + -------------- | \ 2 /|
2 |1 + ------------| *cos(l)
cos (2*l) | 2 |
\ cos (l) /
$$\frac{1 - \frac{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 l \right)}}}{1 + \frac{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 l \right)}}} + \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(l \right)}}\right) \cos{\left(l - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(l - \frac{\pi}{2} \right)}}{\cos^{2}{\left(l \right)}}\right)^{2} \cos{\left(l \right)}}$$
// / 3*pi\ \ // 1 for 2*l mod pi = 0\
// 1 for l mod pi = 0\ || 1 for |2*l + ----| mod 2*pi = 0| || |
|< |*|< \ 2 / | + |< 2 / 2 \ |
\\cos(2*l) otherwise / || | ||sin (2*l)*\-1 + cot (2*l)/ otherwise |
\\sin(2*l) otherwise / \\ /
$$\left(\left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\cot^{2}{\left(2 l \right)} - 1\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
2/pi \ / 2/pi \\
csc |-- - 2*l| | csc |-- - l||
\2 / | \2 /| /pi \
1 - -------------- 2*|1 - ------------|*csc|-- - l|
2 | 2 | \2 /
csc (2*l) \ csc (l) /
------------------ + --------------------------------
2/pi \ 2
csc |-- - 2*l| / 2/pi \\
\2 / | csc |-- - l||
1 + -------------- | \2 /|
2 |1 + ------------| *csc(l)
csc (2*l) | 2 |
\ csc (l) /
$$\frac{1 - \frac{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 l \right)}}}{1 + \frac{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 l \right)}}} + \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(l \right)}}\right) \csc{\left(- l + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}}{\csc^{2}{\left(l \right)}}\right)^{2} \csc{\left(l \right)}}$$
// 1 for 2*l mod pi = 0\
|| |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || 4 2 / 1 \ |
|< |*|< | + |<4*cos (l)*tan (l)*|-1 + ---------| otherwise |
\\sin(2*l) otherwise / \\cos(2*l) otherwise / || | 2 | |
|| \ tan (2*l)/ |
\\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 l \right)}}\right) \cos^{4}{\left(l \right)} \tan^{2}{\left(l \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || / 2 \ |
|< |*|< | + |< 2 | sin (4*l) | |
\\sin(2*l) otherwise / \\cos(2*l) otherwise / ||sin (2*l)*|-1 + -----------| otherwise |
|| | 4 | |
\\ \ 4*sin (2*l)/ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 l \right)}}{4 \sin^{4}{\left(2 l \right)}}\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
// 1 for l mod pi = 0\ || |
// 0 for 2*l mod pi = 0\ || | || / 2 \ |
|< |*|< /pi \ | + |< 2 | sin (4*l) | |
\\sin(2*l) otherwise / ||sin|-- + 2*l| otherwise | ||sin (2*l)*|-1 + -----------| otherwise |
\\ \2 / / || | 4 | |
\\ \ 4*sin (2*l)/ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\sin{\left(2 l + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 l \right)}}{4 \sin^{4}{\left(2 l \right)}}\right) \sin^{2}{\left(2 l \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
|| / 2 \ |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || | sin (4*l) | |
|< |*|< | + |<(1 - cos(4*l))*|-1 + -----------| |
\\sin(2*l) otherwise / \\cos(2*l) otherwise / || | 4 | |
|| \ 4*sin (2*l)/ |
||--------------------------------- otherwise |
\\ 2 /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(4 l \right)}}{4 \sin^{4}{\left(2 l \right)}}\right) \left(- \cos{\left(4 l \right)} + 1\right)}{2} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\ || | || | || | || 2*cot(l) | || 2 | || 2 | |<----------- otherwise |*|<-1 + cot (l) | + |<-1 + cot (2*l) | || 2 | ||------------ otherwise | ||-------------- otherwise | ||1 + cot (l) | || 2 | || 2 | \\ / \\1 + cot (l) / \\1 + cot (2*l) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\cot^{2}{\left(l \right)} - 1}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 l \right)} - 1}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
// 0 for 2*l mod pi = 0\ || / 2 \ |
|| | // 1 for l mod pi = 0\ || 2/ pi\ | cos (2*l) | |
|< / pi\ |*|< | + |
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\cos{\left(2 l - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 l \right)}}{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 l - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
// 1 for l mod pi = 0\ || 2 |
// 0 for 2*l mod pi = 0\ || | || csc (2*l) |
|| | || 1 | ||-1 + -------------- |
|< 1 |*|<------------- otherwise | + |< 2/pi \ |
||-------- otherwise | || /pi \ | || csc |-- - 2*l| |
\\csc(2*l) / ||csc|-- - 2*l| | || \2 / |
\\ \2 / / ||------------------- otherwise |
|| 2 |
\\ csc (2*l) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{1}{\csc{\left(2 l \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 l + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 l \right)}}{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 l \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
|| 2/ pi\ |
// 0 for 2*l mod pi = 0\ || sec |2*l - --| |
|| | // 1 for l mod pi = 0\ || \ 2 / |
|| 1 | || | ||-1 + -------------- |
|<------------- otherwise |*|< 1 | + |< 2 |
|| / pi\ | ||-------- otherwise | || sec (2*l) |
||sec|2*l - --| | \\sec(2*l) / ||------------------- otherwise |
\\ \ 2 / / || 2/ pi\ |
|| sec |2*l - --| |
|| \ 2 / |
\\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{1}{\sec{\left(2 l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{1}{\sec{\left(2 l \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 l \right)}}}{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\
// 0 for 2*l mod pi = 0\ || | || |
|| | || 1 | || 1 |
|| 2 | ||-1 + ------- | ||-1 + --------- |
||-------------------- otherwise | || 2 | || 2 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(l \right)}}\right) \tan{\left(l \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(l \right)}}}{1 + \frac{1}{\tan^{2}{\left(l \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 l \right)}}}{1 + \frac{1}{\tan^{2}{\left(2 l \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
|| |
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || 2 / 1 \ |
|| | || | ||4*tan (l)*|-1 + ---------| |
|| 2*tan(l) | || 2 | || | 2 | |
|<----------- otherwise |*|<1 - tan (l) | + |< \ tan (2*l)/ |
|| 2 | ||----------- otherwise | ||-------------------------- otherwise |
||1 + tan (l) | || 2 | || 2 |
\\ / \\1 + tan (l) / || / 2 \ |
|| \1 + tan (l)/ |
\\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \tan{\left(l \right)}}{\tan^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{- \tan^{2}{\left(l \right)} + 1}{\tan^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 l \right)}}\right) \tan^{2}{\left(l \right)}}{\left(\tan^{2}{\left(l \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |2*l + ----| mod 2*pi = 0|
// 1 for l mod pi = 0\ || \ 2 / | // 1 for 2*l mod pi = 0\
|| | || | || |
|| 2 | || 2/ pi\ | || 2 |
|<-1 + cot (l) |*|<-1 + tan |l + --| | + |<-1 + cot (2*l) |
||------------ otherwise | || \ 4 / | ||-------------- otherwise |
|| 2 | ||----------------- otherwise | || 2 |
\\1 + cot (l) / || 2/ pi\ | \\1 + cot (2*l) /
|| 1 + tan |l + --| |
\\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\cot^{2}{\left(l \right)} - 1}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 l + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(l + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(l + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 l \right)} - 1}{\cot^{2}{\left(2 l \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 2*l| mod pi = 0| || 0 for |-- + 4*l| mod pi = 0|
// 0 for 2*l mod pi = 0\ || \2 / | || \2 / |
|| | || | || |
|| 2*cot(l) | || / pi\ | || / pi\ |
|<----------- otherwise |*|< 2*cot|l + --| | + |< 2*cot|2*l + --| |
|| 2 | || \ 4 / | || \ 4 / |
||1 + cot (l) | ||---------------- otherwise | ||------------------ otherwise |
\\ / || 2/ pi\ | || 2/ pi\ |
||1 + cot |l + --| | ||1 + cot |2*l + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(l + \frac{\pi}{4} \right)}}{\cot^{2}{\left(l + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(4 l + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 l + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 l + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || |
|| | || | || // 0 for 2*l mod pi = 0\ |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\sin{\left(2 l \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\cos{\left(2 l \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\cot^{2}{\left(2 l \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{- \cos{\left(4 l \right)} + 1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\
|| | || |
// 0 for 2*l mod pi = 0\ || 2 | || 2 |
|| | || sin (2*l) | || sin (4*l) |
|| sin(2*l) | ||-1 + --------- | ||-1 + ----------- |
||----------------------- otherwise | || 4 | || 4 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\sin{\left(2 l \right)}}{\left(1 + \frac{\sin^{2}{\left(2 l \right)}}{4 \sin^{4}{\left(l \right)}}\right) \sin^{2}{\left(l \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 l \right)}}{4 \sin^{4}{\left(l \right)}}}{1 + \frac{\sin^{2}{\left(2 l \right)}}{4 \sin^{4}{\left(l \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(4 l \right)}}{4 \sin^{4}{\left(2 l \right)}}}{1 + \frac{\sin^{2}{\left(4 l \right)}}{4 \sin^{4}{\left(2 l \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\
|| | || |
// 0 for 2*l mod pi = 0\ || 2 | || 2 |
|| | || cos (l) | || cos (2*l) |
|| 2*cos(l) | ||-1 + ------------ | ||-1 + -------------- |
||------------------------------ otherwise | || 2/ pi\ | || 2/ pi\ |
||/ 2 \ | || cos |l - --| | || cos |2*l - --| |
|<| cos (l) | / pi\ |*|< \ 2 / | + |< \ 2 / |
|||1 + ------------|*cos|l - --| | ||----------------- otherwise | ||------------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 | || 2 |
||| cos |l - --|| | || cos (l) | || cos (2*l) |
||\ \ 2 // | || 1 + ------------ | || 1 + -------------- |
\\ / || 2/ pi\ | || 2/ pi\ |
|| cos |l - --| | || cos |2*l - --| |
\\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cos{\left(l \right)}}{\left(\frac{\cos^{2}{\left(l \right)}}{\cos^{2}{\left(l - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(l - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(l \right)}}{\cos^{2}{\left(l - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(l \right)}}{\cos^{2}{\left(l - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(2 l \right)}}{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(2 l \right)}}{\cos^{2}{\left(2 l - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\
// 0 for 2*l mod pi = 0\ || | || |
|| | || 2/ pi\ | || 2/ pi\ |
|| / pi\ | || sec |l - --| | || sec |2*l - --| |
|| 2*sec|l - --| | || \ 2 / | || \ 2 / |
|| \ 2 / | ||-1 + ------------ | ||-1 + -------------- |
||------------------------- otherwise | || 2 | || 2 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \sec{\left(l - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(l \right)}}\right) \sec{\left(l \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(l \right)}}}{1 + \frac{\sec^{2}{\left(l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(l \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 l \right)}}}{1 + \frac{\sec^{2}{\left(2 l - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 l \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*l mod pi = 0\
// 0 for 2*l mod pi = 0\ // 1 for l mod pi = 0\ || |
|| | || | || // 0 for 2*l mod pi = 0\ |
||/ 0 for 2*l mod pi = 0 | ||/ 1 for l mod pi = 0 | || || | |
||| | ||| | || || 2 | |
|<| 2*cot(l) |*|<| 2 | + |
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \cot{\left(l \right)}}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\cot^{2}{\left(l \right)} - 1}{\cot^{2}{\left(l \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\left(\cot^{2}{\left(2 l \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(l \right)}}{\left(\cot^{2}{\left(l \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 1 for l mod pi = 0\ // 1 for 2*l mod pi = 0\
|| | || |
// 0 for 2*l mod pi = 0\ || 2 | || 2 |
|| | || csc (l) | || csc (2*l) |
|| 2*csc(l) | ||-1 + ------------ | ||-1 + -------------- |
||------------------------------ otherwise | || 2/pi \ | || 2/pi \ |
||/ 2 \ | || csc |-- - l| | || csc |-- - 2*l| |
|<| csc (l) | /pi \ |*|< \2 / | + |< \2 / |
|||1 + ------------|*csc|-- - l| | ||----------------- otherwise | ||------------------- otherwise |
||| 2/pi \| \2 / | || 2 | || 2 |
||| csc |-- - l|| | || csc (l) | || csc (2*l) |
||\ \2 // | || 1 + ------------ | || 1 + -------------- |
\\ / || 2/pi \ | || 2/pi \ |
|| csc |-- - l| | || csc |-- - 2*l| |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{2 \csc{\left(l \right)}}{\left(\frac{\csc^{2}{\left(l \right)}}{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- l + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: l \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(l \right)}}{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(l \right)}}{\csc^{2}{\left(- l + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: 2 l \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 l \right)}}{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(2 l \right)}}{\csc^{2}{\left(- 2 l + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
Piecewise((0, Mod(2*l = pi, 0)), (2*csc(l)/((1 + csc(l)^2/csc(pi/2 - l)^2)*csc(pi/2 - l)), True))*Piecewise((1, Mod(l = pi, 0)), ((-1 + csc(l)^2/csc(pi/2 - l)^2)/(1 + csc(l)^2/csc(pi/2 - l)^2), True)) + Piecewise((1, Mod(2*l = pi, 0)), ((-1 + csc(2*l)^2/csc(pi/2 - 2*l)^2)/(1 + csc(2*l)^2/csc(pi/2 - 2*l)^2), True))