Господин Экзамен
sin(2*d)+cos(2*d)+tan(2*d) если d=-1/4
Решение
Вы ввели
[src]
sin(2*d) + cos(2*d) + tan(2*d)
$$\sin{\left(2 d \right)} + \cos{\left(2 d \right)} + \tan{\left(2 d \right)}$$
sin(2*d) + cos(2*d) + tan(2*d)
Общее упрощение
[src]
___ / pi\
\/ 2 *sin|2*d + --| + tan(2*d)
\ 4 /
$$\sqrt{2} \sin{\left(2 d + \frac{\pi}{4} \right)} + \tan{\left(2 d \right)}$$
sqrt(2)*sin(2*d + pi/4) + tan(2*d)
Подстановка условия
[src]
sin(2*d) + cos(2*d) + tan(2*d) при d = -1/4
подставляем
sin(2*d) + cos(2*d) + tan(2*d)
$$\sin{\left(2 d \right)} + \cos{\left(2 d \right)} + \tan{\left(2 d \right)}$$
___ / pi\
\/ 2 *sin|2*d + --| + tan(2*d)
\ 4 /
$$\sqrt{2} \sin{\left(2 d + \frac{\pi}{4} \right)} + \tan{\left(2 d \right)}$$
переменные
d = -1/4
$$d = - \frac{1}{4}$$
___ / pi\
\/ 2 *sin|2*(-1/4) + --| + tan(2*(-1/4))
\ 4 /
$$\sqrt{2} \sin{\left(2 (-1/4) + \frac{\pi}{4} \right)} + \tan{\left(2 (-1/4) \right)}$$
___ /1 pi\
-tan(1/2) + \/ 2 *cos|- + --|
\2 4 /
$$- \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \cos{\left(\frac{1}{2} + \frac{\pi}{4} \right)}$$
-tan(1/2) + sqrt(2)*cos(1/2 + pi/4)
Степени
[src]
-2*I*d 2*I*d / -2*I*d 2*I*d\ / 2*I*d -2*I*d\
e e I*\- e + e / I*\- e + e /
------- + ------ - ---------------------- + ----------------------
2 2 2 -2*I*d 2*I*d
e + e
$$\frac{e^{2 i d}}{2} - \frac{i \left(e^{2 i d} - e^{- 2 i d}\right)}{2} + \frac{i \left(- e^{2 i d} + e^{- 2 i d}\right)}{e^{2 i d} + e^{- 2 i d}} + \frac{e^{- 2 i d}}{2}$$
exp(-2*i*d)/2 + exp(2*i*d)/2 - i*(-exp(-2*i*d) + exp(2*i*d))/2 + i*(-exp(2*i*d) + exp(-2*i*d))/(exp(-2*i*d) + exp(2*i*d))
Тригонометрическая часть
[src]
___ / pi\
\/ 2 *sin|2*d + --| + tan(2*d)
\ 4 /
$$\sqrt{2} \sin{\left(2 d + \frac{\pi}{4} \right)} + \tan{\left(2 d \right)}$$
___ / pi\
\/ 2 *cos|2*d - --| + tan(2*d)
\ 4 /
$$\sqrt{2} \cos{\left(2 d - \frac{\pi}{4} \right)} + \tan{\left(2 d \right)}$$
2 -1 + 2*cos (d) + sin(2*d) + tan(2*d)
$$2 \cos^{2}{\left(d \right)} + \sin{\left(2 d \right)} + \tan{\left(2 d \right)} - 1$$
sin(2*d) -------- + cos(2*d) + sin(2*d) cos(2*d)
$$\sin{\left(2 d \right)} + \cos{\left(2 d \right)} + \frac{\sin{\left(2 d \right)}}{\cos{\left(2 d \right)}}$$
2
___ / pi\ 2*sin (2*d)
\/ 2 *sin|2*d + --| + -----------
\ 4 / sin(4*d)
$$\frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}} + \sqrt{2} \sin{\left(2 d + \frac{\pi}{4} \right)}$$
2 2*sin (2*d) ----------- + cos(2*d) + sin(2*d) sin(4*d)
$$\frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}} + \sin{\left(2 d \right)} + \cos{\left(2 d \right)}$$
/ pi\
cos|2*d - --|
___ / pi\ \ 2 /
\/ 2 *cos|2*d - --| + -------------
\ 4 / cos(2*d)
$$\sqrt{2} \cos{\left(2 d - \frac{\pi}{4} \right)} + \frac{\cos{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(2 d \right)}}$$
1 1 sec(2*d) -------- + -------- + -------- csc(2*d) sec(2*d) csc(2*d)
$$\frac{\sec{\left(2 d \right)}}{\csc{\left(2 d \right)}} + \frac{1}{\sec{\left(2 d \right)}} + \frac{1}{\csc{\left(2 d \right)}}$$
___ / pi\
2*\/ 2 *tan|d + --|
\ 8 /
------------------- + tan(2*d)
2/ pi\
1 + tan |d + --|
\ 8 /
$$\tan{\left(2 d \right)} + \frac{2 \sqrt{2} \tan{\left(d + \frac{\pi}{8} \right)}}{\tan^{2}{\left(d + \frac{\pi}{8} \right)} + 1}$$
2 2 cos (d) - sin (d) + 2*cos(d)*sin(d) + tan(2*d)
$$- \sin^{2}{\left(d \right)} + 2 \sin{\left(d \right)} \cos{\left(d \right)} + \cos^{2}{\left(d \right)} + \tan{\left(2 d \right)}$$
2 2*sin (2*d) /pi \ ----------- + sin(2*d) + sin|-- + 2*d| sin(4*d) \2 /
$$\frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}} + \sin{\left(2 d \right)} + \sin{\left(2 d + \frac{\pi}{2} \right)}$$
___
\/ 2 sec(2*d)
------------- + -------------
/ pi\ / pi\
sec|2*d - --| sec|2*d - --|
\ 4 / \ 2 /
$$\frac{\sec{\left(2 d \right)}}{\sec{\left(2 d - \frac{\pi}{2} \right)}} + \frac{\sqrt{2}}{\sec{\left(2 d - \frac{\pi}{4} \right)}}$$
/pi \
___ csc|-- - 2*d|
\/ 2 \2 /
------------- + -------------
/ pi\ csc(2*d)
csc|2*d + --|
\ 4 /
$$\frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(2 d \right)}} + \frac{\sqrt{2}}{\csc{\left(2 d + \frac{\pi}{4} \right)}}$$
1 1 2*csc(4*d)
-------- + -------- + ----------
csc(2*d) sec(2*d) 2
csc (2*d)
$$\frac{1}{\sec{\left(2 d \right)}} + \frac{1}{\csc{\left(2 d \right)}} + \frac{2 \csc{\left(4 d \right)}}{\csc^{2}{\left(2 d \right)}}$$
sin(2*d) /pi \ ------------- + sin(2*d) + sin|-- + 2*d| /pi \ \2 / sin|-- + 2*d| \2 /
$$\sin{\left(2 d \right)} + \sin{\left(2 d + \frac{\pi}{2} \right)} + \frac{\sin{\left(2 d \right)}}{\sin{\left(2 d + \frac{\pi}{2} \right)}}$$
/ pi\ cos|2*d - --| \ 2 / / pi\ ------------- + cos(2*d) + cos|2*d - --| cos(2*d) \ 2 /
$$\cos{\left(2 d \right)} + \cos{\left(2 d - \frac{\pi}{2} \right)} + \frac{\cos{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(2 d \right)}}$$
2 ___ 2 / pi\
1 - 2*cos (d) + 2*\/ 2 *cos (d)*sin|2*d + --|
\ 4 /
---------------------------------------------
cos(2*d)
$$\frac{2 \sqrt{2} \sin{\left(2 d + \frac{\pi}{4} \right)} \cos^{2}{\left(d \right)} - 2 \cos^{2}{\left(d \right)} + 1}{\cos{\left(2 d \right)}}$$
2
1 - tan (d) 2*tan(d)
----------- + ----------- + tan(2*d)
2 2
1 + tan (d) 1 + tan (d)
$$\tan{\left(2 d \right)} + \frac{- \tan^{2}{\left(d \right)} + 1}{\tan^{2}{\left(d \right)} + 1} + \frac{2 \tan{\left(d \right)}}{\tan^{2}{\left(d \right)} + 1}$$
1 1 sec(2*d)
-------- + ------------- + -------------
sec(2*d) / pi\ / pi\
sec|2*d - --| sec|2*d - --|
\ 2 / \ 2 /
$$\frac{\sec{\left(2 d \right)}}{\sec{\left(2 d - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 d - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 d \right)}}$$
2/ pi\
2*cos |2*d - --|
\ 2 / / pi\
---------------- + cos(2*d) + cos|2*d - --|
/ pi\ \ 2 /
cos|4*d - --|
\ 2 /
$$\cos{\left(2 d \right)} + \frac{2 \cos^{2}{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(4 d - \frac{\pi}{2} \right)}} + \cos{\left(2 d - \frac{\pi}{2} \right)}$$
/pi \
csc|-- - 2*d|
1 1 \2 /
-------- + ------------- + -------------
csc(2*d) /pi \ csc(2*d)
csc|-- - 2*d|
\2 /
$$\frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(2 d \right)}} + \frac{1}{\csc{\left(- 2 d + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 d \right)}}$$
1 1 sec(2*d)
-------- + ------------- + -------------
sec(2*d) /pi \ /pi \
sec|-- - 2*d| sec|-- - 2*d|
\2 / \2 /
$$\frac{\sec{\left(2 d \right)}}{\sec{\left(- 2 d + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- 2 d + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 d \right)}}$$
/pi \
csc|-- - 2*d|
1 1 \2 /
------------- + ------------- + -------------
csc(pi - 2*d) /pi \ csc(pi - 2*d)
csc|-- - 2*d|
\2 /
$$\frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(- 2 d + \pi \right)}} + \frac{1}{\csc{\left(- 2 d + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 d + \pi \right)}}$$
/ pi\
2*sec|4*d - --|
1 1 \ 2 /
-------- + ------------- + ---------------
sec(2*d) / pi\ 2/ pi\
sec|2*d - --| sec |2*d - --|
\ 2 / \ 2 /
$$\frac{1}{\sec{\left(2 d - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(4 d - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 d - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 d \right)}}$$
2
1 - tan (d) 2*tan(d) 2*tan(d)
----------- + ----------- + -----------
2 2 2
1 + tan (d) 1 + tan (d) 1 - tan (d)
$$\frac{- \tan^{2}{\left(d \right)} + 1}{\tan^{2}{\left(d \right)} + 1} + \frac{2 \tan{\left(d \right)}}{\tan^{2}{\left(d \right)} + 1} + \frac{2 \tan{\left(d \right)}}{- \tan^{2}{\left(d \right)} + 1}$$
1
1 - -------
2
1 cot (d) 2
-------- + ----------- + --------------------
cot(2*d) 1 / 1 \
1 + ------- |1 + -------|*cot(d)
2 | 2 |
cot (d) \ cot (d)/
$$\frac{1 - \frac{1}{\cot^{2}{\left(d \right)}}}{1 + \frac{1}{\cot^{2}{\left(d \right)}}} + \frac{1}{\cot{\left(2 d \right)}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(d \right)}}\right) \cot{\left(d \right)}}$$
// / pi\ \
|| 0 for |2*d + --| mod pi = 0|
___ || \ 4 / |
\/ 2 *|< | + tan(2*d)
|| 2/ pi\ / pi\ |
||2*sin |d + --|*cot|d + --| otherwise |
\\ \ 8 / \ 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 d + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(d + \frac{\pi}{8} \right)} \cot{\left(d + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + \tan{\left(2 d \right)}$$
// 0 for 2*d mod pi = 0\ // 1 for d mod pi = 0\ |< | + |< | + tan(2*d) \\sin(2*d) otherwise / \\cos(2*d) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\cos{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \tan{\left(2 d \right)}$$
// / pi\ \
|| 0 for |2*d + --| mod pi = 0|
|| \ 4 / |
|| |
1 ___ || / pi\ |
-------- + \/ 2 *|< 2*cot|d + --| |
cot(2*d) || \ 8 / |
||---------------- otherwise |
|| 2/ pi\ |
||1 + cot |d + --| |
\\ \ 8 / /
$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 d + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(d + \frac{\pi}{8} \right)}}{\cot^{2}{\left(d + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \frac{1}{\cot{\left(2 d \right)}}$$
/ pi\
2*tan|d + --| 2 / 2 \
2*tan(d) \ 4 / 4*tan (d)*\1 + tan (2*d)/
----------- + ---------------- + -------------------------
2 2/ pi\ 2
1 + tan (d) 1 + tan |d + --| / 2 \
\ 4 / \1 + tan (d)/ *tan(2*d)
$$\frac{2 \tan{\left(d + \frac{\pi}{4} \right)}}{\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(d \right)}}{\tan^{2}{\left(d \right)} + 1} + \frac{4 \left(\tan^{2}{\left(2 d \right)} + 1\right) \tan^{2}{\left(d \right)}}{\left(\tan^{2}{\left(d \right)} + 1\right)^{2} \tan{\left(2 d \right)}}$$
2 // 1 for d mod pi = 0\
2*sin (2*d) // 0 for 2*d mod pi = 0\ || |
----------- + |< | + |< /pi \ |
sin(4*d) \\sin(2*d) otherwise / ||sin|-- + 2*d| otherwise |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\sin{\left(2 d + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}}$$
/ pi\
cos|2*d - --| // 0 for 2*d mod pi = 0\
\ 2 / || | // 1 for d mod pi = 0\
------------- + |< / pi\ | + |< |
cos(2*d) ||cos|2*d - --| otherwise | \\cos(2*d) otherwise /
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\cos{\left(2 d - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\cos{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \frac{\cos{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(2 d \right)}}$$
/ pi\ / 2/ pi\\
2*tan|d + --| |1 + tan |d + --||*cot(d)
2*cot(d) \ 4 / \ \ 4 //
----------- + ---------------- + -------------------------
2 2/ pi\ / 2 \ / pi\
1 + cot (d) 1 + tan |d + --| \1 + cot (d)/*tan|d + --|
\ 4 / \ 4 /
$$\frac{\left(\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1\right) \cot{\left(d \right)}}{\left(\cot^{2}{\left(d \right)} + 1\right) \tan{\left(d + \frac{\pi}{4} \right)}} + \frac{2 \cot{\left(d \right)}}{\cot^{2}{\left(d \right)} + 1} + \frac{2 \tan{\left(d + \frac{\pi}{4} \right)}}{\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1}$$
// 0 for 2*d mod pi = 0\ // 1 for d mod pi = 0\ || | || | || 2*tan(d) | || 2 | |<----------- otherwise | + |<1 - tan (d) | + tan(2*d) || 2 | ||----------- otherwise | ||1 + tan (d) | || 2 | \\ / \\1 + tan (d) /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \tan{\left(d \right)}}{\tan^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{- \tan^{2}{\left(d \right)} + 1}{\tan^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \tan{\left(2 d \right)}$$
// 0 for 2*d mod pi = 0\
|| | // 1 for d mod pi = 0\
sec(2*d) || 1 | || |
------------- + |<------------- otherwise | + |< 1 |
/ pi\ || / pi\ | ||-------- otherwise |
sec|2*d - --| ||sec|2*d - --| | \\sec(2*d) /
\ 2 / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{1}{\sec{\left(2 d - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{1}{\sec{\left(2 d \right)}} & \text{otherwise} \end{cases}\right) + \frac{\sec{\left(2 d \right)}}{\sec{\left(2 d - \frac{\pi}{2} \right)}}$$
/pi \ // 1 for d mod pi = 0\
csc|-- - 2*d| // 0 for 2*d mod pi = 0\ || |
\2 / || | || 1 |
------------- + |< 1 | + |<------------- otherwise |
csc(2*d) ||-------- otherwise | || /pi \ |
\\csc(2*d) / ||csc|-- - 2*d| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{1}{\csc{\left(2 d \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 d + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(2 d \right)}}$$
4
4*sin (d)
1 - ---------
2 2 2
sin (2*d) 2*sin (2*d) 4*sin (d)
------------- + ----------- + ------------------------
4 sin(4*d) / 4 \
4*sin (d) | 4*sin (d)|
1 + --------- |1 + ---------|*sin(2*d)
2 | 2 |
sin (2*d) \ sin (2*d)/
$$\frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}} + \frac{- \frac{4 \sin^{4}{\left(d \right)}}{\sin^{2}{\left(2 d \right)}} + 1}{\frac{4 \sin^{4}{\left(d \right)}}{\sin^{2}{\left(2 d \right)}} + 1} + \frac{4 \sin^{2}{\left(d \right)}}{\left(\frac{4 \sin^{4}{\left(d \right)}}{\sin^{2}{\left(2 d \right)}} + 1\right) \sin{\left(2 d \right)}}$$
// 0 for 2*d mod pi = 0\ // 1 for d mod pi = 0\
|| | || |
1 || 2*cot(d) | || 2 |
-------- + |<----------- otherwise | + |<-1 + cot (d) |
cot(2*d) || 2 | ||------------ otherwise |
||1 + cot (d) | || 2 |
\\ / \\1 + cot (d) /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \cot{\left(d \right)}}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\cot^{2}{\left(d \right)} - 1}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \frac{1}{\cot{\left(2 d \right)}}$$
2/ pi\ / 2 \ / 2/ pi\\
2 -1 + tan |d + --| \1 + cot (d)/*|-1 + tan |d + --||
-1 + cot (d) \ 4 / \ \ 4 //
------------ + ----------------- + ---------------------------------
2 2/ pi\ / 2/ pi\\ / 2 \
1 + cot (d) 1 + tan |d + --| |1 + tan |d + --||*\-1 + cot (d)/
\ 4 / \ \ 4 //
$$\frac{\tan^{2}{\left(d + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1} + \frac{\left(\tan^{2}{\left(d + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(d \right)} + 1\right)}{\left(\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(d \right)} - 1\right)} + \frac{\cot^{2}{\left(d \right)} - 1}{\cot^{2}{\left(d \right)} + 1}$$
// 1 for d mod pi = 0\
// 0 for 2*d mod pi = 0\ || |
|| | || 1 |
|| 2 | ||-1 + ------- |
||-------------------- otherwise | || 2 |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(d \right)}}\right) \tan{\left(d \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(d \right)}}}{1 + \frac{1}{\tan^{2}{\left(d \right)}}} & \text{otherwise} \end{cases}\right) + \tan{\left(2 d \right)}$$
2/ pi\ / 2 \ / 2/ pi\\
1 - cot |d + --| 2 \1 + tan (d)/*|1 - cot |d + --||
\ 4 / 1 - tan (d) \ \ 4 //
---------------- + ----------- + --------------------------------
2/ pi\ 2 / 2/ pi\\ / 2 \
1 + cot |d + --| 1 + tan (d) |1 + cot |d + --||*\1 - tan (d)/
\ 4 / \ \ 4 //
$$\frac{- \tan^{2}{\left(d \right)} + 1}{\tan^{2}{\left(d \right)} + 1} + \frac{- \cot^{2}{\left(d + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(d + \frac{\pi}{4} \right)} + 1} + \frac{\left(- \cot^{2}{\left(d + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(d \right)} + 1\right)}{\left(- \tan^{2}{\left(d \right)} + 1\right) \left(\cot^{2}{\left(d + \frac{\pi}{4} \right)} + 1\right)}$$
2/ pi\
cos |d - --|
\ 2 /
1 - ------------ / pi\ / pi\
2 cos|2*d - --| 2*cos|d - --|
cos (d) \ 2 / \ 2 /
---------------- + ------------- + -------------------------
2/ pi\ cos(2*d) / 2/ pi\\
cos |d - --| | cos |d - --||
\ 2 / | \ 2 /|
1 + ------------ |1 + ------------|*cos(d)
2 | 2 |
cos (d) \ cos (d) /
$$\frac{1 - \frac{\cos^{2}{\left(d - \frac{\pi}{2} \right)}}{\cos^{2}{\left(d \right)}}}{1 + \frac{\cos^{2}{\left(d - \frac{\pi}{2} \right)}}{\cos^{2}{\left(d \right)}}} + \frac{\cos{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(2 d \right)}} + \frac{2 \cos{\left(d - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(d - \frac{\pi}{2} \right)}}{\cos^{2}{\left(d \right)}}\right) \cos{\left(d \right)}}$$
2
sec (d)
1 - ------------
2/ pi\
sec |d - --|
\ 2 / sec(2*d) 2*sec(d)
---------------- + ------------- + ------------------------------
2 / pi\ / 2 \
sec (d) sec|2*d - --| | sec (d) | / pi\
1 + ------------ \ 2 / |1 + ------------|*sec|d - --|
2/ pi\ | 2/ pi\| \ 2 /
sec |d - --| | sec |d - --||
\ 2 / \ \ 2 //
$$\frac{- \frac{\sec^{2}{\left(d \right)}}{\sec^{2}{\left(d - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(d \right)}}{\sec^{2}{\left(d - \frac{\pi}{2} \right)}} + 1} + \frac{\sec{\left(2 d \right)}}{\sec{\left(2 d - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(d \right)}}{\left(\frac{\sec^{2}{\left(d \right)}}{\sec^{2}{\left(d - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(d - \frac{\pi}{2} \right)}}$$
2/pi \
csc |-- - d|
\2 /
1 - ------------ /pi \ /pi \
2 csc|-- - 2*d| 2*csc|-- - d|
csc (d) \2 / \2 /
---------------- + ------------- + -------------------------
2/pi \ csc(2*d) / 2/pi \\
csc |-- - d| | csc |-- - d||
\2 / | \2 /|
1 + ------------ |1 + ------------|*csc(d)
2 | 2 |
csc (d) \ csc (d) /
$$\frac{1 - \frac{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}}{\csc^{2}{\left(d \right)}}}{1 + \frac{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}}{\csc^{2}{\left(d \right)}}} + \frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(2 d \right)}} + \frac{2 \csc{\left(- d + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}}{\csc^{2}{\left(d \right)}}\right) \csc{\left(d \right)}}$$
// 0 for 2*d mod pi = 0\ // 1 for d mod pi = 0\
|| | || |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\cos{\left(2 d \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \tan{\left(2 d \right)}$$
// 0 for 2*d mod pi = 0\ // 1 for d mod pi = 0\
|| | || |
||/ 0 for 2*d mod pi = 0 | ||/ 1 for d mod pi = 0 |
1 ||| | ||| |
-------- + |<| 2*cot(d) | + |<| 2 |
cot(2*d) ||<----------- otherwise otherwise | ||<-1 + cot (d) otherwise |
||| 2 | |||------------ otherwise |
|||1 + cot (d) | ||| 2 |
\\\ / \\\1 + cot (d) /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \cot{\left(d \right)}}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\cot^{2}{\left(d \right)} - 1}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \frac{1}{\cot{\left(2 d \right)}}$$
// 1 for d mod pi = 0\
|| |
// 0 for 2*d mod pi = 0\ || 2 |
|| | || sin (2*d) |
2 || sin(2*d) | ||-1 + --------- |
2*sin (2*d) ||----------------------- otherwise | || 4 |
----------- + |
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{\sin{\left(2 d \right)}}{\left(1 + \frac{\sin^{2}{\left(2 d \right)}}{4 \sin^{4}{\left(d \right)}}\right) \sin^{2}{\left(d \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 d \right)}}{4 \sin^{4}{\left(d \right)}}}{1 + \frac{\sin^{2}{\left(2 d \right)}}{4 \sin^{4}{\left(d \right)}}} & \text{otherwise} \end{cases}\right) + \frac{2 \sin^{2}{\left(2 d \right)}}{\sin{\left(4 d \right)}}$$
// 0 for 2*d mod pi = 0\ // zoo for 4*d mod pi = 0\ // /pi \ \ || | || | // 0 for 2*d mod pi = 0\ || 0 for |-- + 2*d| mod pi = 0| 2*|<1 - cos(4*d) |*|< 1 | + |< | + |< \2 / | ||------------ otherwise | ||-------- otherwise | \\sin(2*d) otherwise / || | \\ 2 / \\sin(4*d) / \\cos(2*d) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{- \cos{\left(4 d \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 4 d \bmod \pi = 0 \\\frac{1}{\sin{\left(4 d \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 d + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 d \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for d mod pi = 0\ // / 3*pi\ \ // / 3*pi\ \ || | || 1 for |2*d + ----| mod 2*pi = 0| // 1 for d mod pi = 0\ || 1 for |2*d + ----| mod 2*pi = 0| |< 1 |*|< \ 2 / | + |< | + |< \ 2 / | ||-------- otherwise | || | \\cos(2*d) otherwise / || | \\cos(2*d) / \\sin(2*d) otherwise / \\sin(2*d) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{1}{\cos{\left(2 d \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 d + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\cos{\left(2 d \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 d + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 d \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for d mod pi = 0\
|| |
// 0 for 2*d mod pi = 0\ || 2 |
|| | || cos (d) |
/ pi\ || 2*cos(d) | ||-1 + ------------ |
cos|2*d - --| ||------------------------------ otherwise | || 2/ pi\ |
\ 2 / ||/ 2 \ | || cos |d - --| |
------------- + |<| cos (d) | / pi\ | + |< \ 2 / |
cos(2*d) |||1 + ------------|*cos|d - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 |
||| cos |d - --|| | || cos (d) |
||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ |
|| cos |d - --| |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \cos{\left(d \right)}}{\left(\frac{\cos^{2}{\left(d \right)}}{\cos^{2}{\left(d - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(d - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(d \right)}}{\cos^{2}{\left(d - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(d \right)}}{\cos^{2}{\left(d - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \frac{\cos{\left(2 d - \frac{\pi}{2} \right)}}{\cos{\left(2 d \right)}}$$
// 1 for d mod pi = 0\
// 0 for 2*d mod pi = 0\ || |
|| | || 2/ pi\ |
|| / pi\ | || sec |d - --| |
|| 2*sec|d - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ |
sec(2*d) ||------------------------- otherwise | || 2 |
------------- + |
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \sec{\left(d - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(d - \frac{\pi}{2} \right)}}{\sec^{2}{\left(d \right)}}\right) \sec{\left(d \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(d - \frac{\pi}{2} \right)}}{\sec^{2}{\left(d \right)}}}{1 + \frac{\sec^{2}{\left(d - \frac{\pi}{2} \right)}}{\sec^{2}{\left(d \right)}}} & \text{otherwise} \end{cases}\right) + \frac{\sec{\left(2 d \right)}}{\sec{\left(2 d - \frac{\pi}{2} \right)}}$$
// 1 for d mod pi = 0\
|| |
// 0 for 2*d mod pi = 0\ || 2 |
|| | || csc (d) |
/pi \ || 2*csc(d) | ||-1 + ------------ |
csc|-- - 2*d| ||------------------------------ otherwise | || 2/pi \ |
\2 / ||/ 2 \ | || csc |-- - d| |
------------- + |<| csc (d) | /pi \ | + |< \2 / |
csc(2*d) |||1 + ------------|*csc|-- - d| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 |
||| csc |-- - d|| | || csc (d) |
||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ |
|| csc |-- - d| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \csc{\left(d \right)}}{\left(\frac{\csc^{2}{\left(d \right)}}{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- d + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(d \right)}}{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(d \right)}}{\csc^{2}{\left(- d + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \frac{\csc{\left(- 2 d + \frac{\pi}{2} \right)}}{\csc{\left(2 d \right)}}$$
// /pi \ \
// 0 for 2*d mod pi = 0\ || 0 for |-- + 2*d| mod pi = 0|
|| | // zoo for 4*d mod pi = 0\ // 0 for 2*d mod pi = 0\ || \2 / |
|| 2 | || | || | || |
|| 4*cot (d) | || 2 | || 2*cot(d) | || / pi\ |
2*|<-------------- otherwise |*|<1 + cot (2*d) | + |<----------- otherwise | + |< 2*cot|d + --| |
|| 2 | ||------------- otherwise | || 2 | || \ 4 / |
||/ 2 \ | || 2*cot(2*d) | ||1 + cot (d) | ||---------------- otherwise |
||\1 + cot (d)/ | \\ / \\ / || 2/ pi\ |
\\ / ||1 + cot |d + --| |
\\ \ 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(d \right)}}{\left(\cot^{2}{\left(d \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 4 d \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 d \right)} + 1}{2 \cot{\left(2 d \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 2 d \bmod \pi = 0 \\\frac{2 \cot{\left(d \right)}}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 d + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(d + \frac{\pi}{4} \right)}}{\cot^{2}{\left(d + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |2*d + ----| mod 2*pi = 0| || 1 for |2*d + ----| mod 2*pi = 0|
// 1 for d mod pi = 0\ || \ 2 / | // 1 for d mod pi = 0\ || \ 2 / |
|| | || | || | || |
|| 2 | || 2/ pi\ | || 2 | || 2/ pi\ |
|<1 + cot (d) |*|<-1 + tan |d + --| | + |<-1 + cot (d) | + |<-1 + tan |d + --| |
||------------ otherwise | || \ 4 / | ||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise | || 2 | ||----------------- otherwise |
\\-1 + cot (d) / || 2/ pi\ | \\1 + cot (d) / || 2/ pi\ |
|| 1 + tan |d + --| | || 1 + tan |d + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\cot^{2}{\left(d \right)} + 1}{\cot^{2}{\left(d \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 d + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(d + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: d \bmod \pi = 0 \\\frac{\cot^{2}{\left(d \right)} - 1}{\cot^{2}{\left(d \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(2 d + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(d + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(d + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
Piecewise((1, Mod(d = pi, 0)), ((1 + cot(d)^2)/(-1 + cot(d)^2), True))*Piecewise((1, Mod(2*d + 3*pi/2 = 2*pi, 0)), ((-1 + tan(d + pi/4)^2)/(1 + tan(d + pi/4)^2), True)) + Piecewise((1, Mod(d = pi, 0)), ((-1 + cot(d)^2)/(1 + cot(d)^2), True)) + Piecewise((1, Mod(2*d + 3*pi/2 = 2*pi, 0)), ((-1 + tan(d + pi/4)^2)/(1 + tan(d + pi/4)^2), True))