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