Господин Экзамен
16-6*sin(x)^2-6*cos(x)^2 если x=-3/2
Решение
Вы ввели
[src]
2 2 16 - 6*sin (x) - 6*cos (x)
$$- 6 \sin^{2}{\left(x \right)} - 6 \cos^{2}{\left(x \right)} + 16$$
16 - 6*sin(x)^2 - 6*cos(x)^2
Степени
[src]
2 2
/ I*x -I*x\ / -I*x I*x\
|e e | 3*\- e + e /
16 - 6*|---- + -----| + -------------------
\ 2 2 / 2
$$- 6 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + \frac{3 \left(e^{i x} - e^{- i x}\right)^{2}}{2} + 16$$
16 - 6*(exp(i*x)/2 + exp(-i*x)/2)^2 + 3*(-exp(-i*x) + exp(i*x))^2/2
Объединение рациональных выражений
[src]
/ 2 2 \ 2*\8 - 3*cos (x) - 3*sin (x)/
$$2 \left(- 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 8\right)$$
2*(8 - 3*cos(x)^2 - 3*sin(x)^2)
Тригонометрическая часть
[src]
10
$$10$$
6 6
16 - ------- - -------
2 2
csc (x) sec (x)
$$16 - \frac{6}{\sec^{2}{\left(x \right)}} - \frac{6}{\csc^{2}{\left(x \right)}}$$
2 2/ pi\
16 - 6*cos (x) - 6*cos |x - --|
\ 2 /
$$- 6 \cos^{2}{\left(x \right)} - 6 \cos^{2}{\left(x - \frac{\pi}{2} \right)} + 16$$
2 2/ pi\
16 - 6*sin (x) - 6*sin |x + --|
\ 2 /
$$- 6 \sin^{2}{\left(x \right)} - 6 \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 16$$
6 6
16 - ------- - ------------
2 2/ pi\
sec (x) sec |x - --|
\ 2 /
$$16 - \frac{6}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{6}{\sec^{2}{\left(x \right)}}$$
6 6
16 - ------- - ------------
2 2/pi \
sec (x) sec |-- - x|
\2 /
$$16 - \frac{6}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{6}{\sec^{2}{\left(x \right)}}$$
6 6
16 - ------- - ------------
2 2/pi \
csc (x) csc |-- - x|
\2 /
$$16 - \frac{6}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{6}{\csc^{2}{\left(x \right)}}$$
6 6
16 - ------------ - ------------
2 2/pi \
csc (pi - x) csc |-- - x|
\2 /
$$16 - \frac{6}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{6}{\csc^{2}{\left(- x + \pi \right)}}$$
2 1 - 3*cos(2*x) + 6*(1 - cos(x)) + 12*cos(x)
$$6 \left(- \cos{\left(x \right)} + 1\right)^{2} + 12 \cos{\left(x \right)} - 3 \cos{\left(2 x \right)} + 1$$
2
/ 2/x pi\\ 2
3*|1 - cot |- + --|| *(1 + sin(x))
\ \2 4 //
16 - 3*(1 + cos(2*x)) - -----------------------------------
2
$$- \frac{3 \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{2} - 3 \left(\cos{\left(2 x \right)} + 1\right) + 16$$
2/x pi\
24*tan |- + --|
\2 4 /
16 - 3*(1 - cos(2*x)) - -------------------
2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 3 \cdot \left(- \cos{\left(2 x \right)} + 1\right) + 16 - \frac{24 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
2/x\ / 2/x\\
24*tan |-| 6*|1 - tan |-||
\2/ \ \2//
16 - -------------- - ----------------
2 2
/ 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$- \frac{6 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 16 - \frac{24 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2/x\ 2/x pi\
24*cot |-| 24*tan |- + --|
\2/ \2 4 /
16 - -------------- - -------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$16 - \frac{24 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{24 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/x\ 2/x pi\
24*tan |-| 24*tan |- + --|
\2/ \2 4 /
16 - -------------- - -------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$16 - \frac{24 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{24 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2
/ 1 \
6*|1 - -------|
| 2/x\|
| cot |-||
24 \ \2//
16 - ---------------------- - ----------------
2 2
/ 1 \ 2/x\ / 1 \
|1 + -------| *cot |-| |1 + -------|
| 2/x\| \2/ | 2/x\|
| cot |-|| | cot |-||
\ \2// \ \2//
$$- \frac{6 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + 16 - \frac{24}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
16 - 6*|< 2 | - 6*|< 2 |
||sin (x) otherwise | ||cos (x) otherwise |
\\ / \\ /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 16$$
2 2
/ 2/x\\ / 2/x pi\\
6*|-1 + cot |-|| 6*|-1 + tan |- + --||
\ \2// \ \2 4 //
16 - ----------------- - ----------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$- \frac{6 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{6 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 16$$
2 2
/ 2/x pi\\ / 2/x\\
6*|1 - cot |- + --|| 6*|1 - tan |-||
\ \2 4 // \ \2//
16 - --------------------- - ----------------
2 2
/ 2/x pi\\ / 2/x\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$- \frac{6 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{6 \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
16 - 6*|< 2/ pi\ | - 6*|< 2 |
||cos |x - --| otherwise | ||cos (x) otherwise |
\\ \ 2 / / \\ /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
16 - 6*|< 2 | - 6*|< 2/ pi\ |
||sin (x) otherwise | ||sin |x + --| otherwise |
\\ / \\ \ 2 / /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
16 - 6*|<------------ otherwise | - 6*|<------- otherwise |
|| 2/ pi\ | || 2 |
||sec |x - --| | ||sec (x) |
\\ \ 2 / / \\ /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
16 - 6*|<------- otherwise | - 6*|<------------ otherwise |
|| 2 | || 2/pi \ |
||csc (x) | ||csc |-- - x| |
\\ / \\ \2 / /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
16 - 6*|< 2 | - 6*|< |
||cos (x) otherwise | || 4/x\ 2/x\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(- 6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 16$$
2
/ 4/x\\
| 4*sin |-||
| \2/|
6*|1 - ---------| 4/x\
| 2 | 96*sin |-|
\ sin (x) / \2/
16 - ------------------ - ------------------------
2 2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (x)
| 2 | | 2 |
\ sin (x) / \ sin (x) /
$$- \frac{6 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}} + 16 - \frac{96 \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}}$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
16 - 6*|< 2 | - 6*|< |
||sin (x) otherwise | || 2 2/x pi\ |
\\ / ||(1 + sin(x)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 |
16 - 6*|<| | - 6*|<| |
||< 2 otherwise | ||< 2 otherwise |
|||sin (x) otherwise | |||cos (x) otherwise |
\\\ / \\\ /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 2/x\ | || 2 |
|| 4*cot |-| | ||/ 2/x\\ |
|| \2/ | |||-1 + cot |-|| |
16 - 6*|<-------------- otherwise | - 6*|<\ \2// |
|| 2 | ||--------------- otherwise |
||/ 2/x\\ | || 2 |
|||1 + cot |-|| | || / 2/x\\ |
||\ \2// | || |1 + cot |-|| |
\\ / \\ \ \2// /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 2/x\ | || 2 |
|| 4*tan |-| | ||/ 2/x\\ |
|| \2/ | |||1 - tan |-|| |
16 - 6*|<-------------- otherwise | - 6*|<\ \2// |
|| 2 | ||-------------- otherwise |
||/ 2/x\\ | || 2 |
|||1 + tan |-|| | ||/ 2/x\\ |
||\ \2// | |||1 + tan |-|| |
\\ / \\\ \2// /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
2
/ 2/x pi\\
| cos |- - --||
| \2 2 /|
6*|1 - ------------|
| 2/x\ | 2/x pi\
| cos |-| | 24*cos |- - --|
\ \2/ / \2 2 /
16 - --------------------- - ---------------------------
2 2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x\
|1 + ------------| |1 + ------------| *cos |-|
| 2/x\ | | 2/x\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$- \frac{6 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + 16 - \frac{24 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\ \
| sec |-| |
| \2/ |
6*|1 - ------------|
| 2/x pi\| 2/x\
| sec |- - --|| 24*sec |-|
\ \2 2 // \2/
16 - --------------------- - --------------------------------
2 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/x pi\| | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$- \frac{6 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + 16 - \frac{24 \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi x\\
| csc |-- - -||
| \2 2/|
6*|1 - ------------|
| 2/x\ | 2/pi x\
| csc |-| | 24*csc |-- - -|
\ \2/ / \2 2/
16 - --------------------- - ---------------------------
2 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/x\
|1 + ------------| |1 + ------------| *csc |-|
| 2/x\ | | 2/x\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$- \frac{6 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + 16 - \frac{24 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/x\| |
|| 2 | ||| tan |-|| |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
|| 2/x\ | || |
|| 4*cot |-| | || 2/x pi\ |
|| \2/ | || 4*cot |- + --| |
16 - 6*|<-------------- otherwise | - 6*|< \2 4 / |
|| 2 | ||------------------- otherwise |
||/ 2/x\\ | || 2 |
|||1 + cot |-|| | ||/ 2/x pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/x\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/x pi\\ |
16 - 6*|<\ \2// | - 6*|<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/x\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/x pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(- 6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
|| 2 | || |
|| sin (x) | || 2 |
||------------------------ otherwise | ||/ 2 4/x\\ |
|| 2 | |||sin (x) - 4*sin |-|| |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (x) | |
|| sin (x) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/x\| |
|| 2 | ||| 4*sin |-|| |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 |
||| | ||| |
||| 2/x\ | ||| 2 |
||| 4*cot |-| | |||/ 2/x\\ |
16 - 6*|<| \2/ | - 6*|<||-1 + cot |-|| |
||<-------------- otherwise otherwise | ||<\ \2// otherwise |
||| 2 | |||--------------- otherwise |
|||/ 2/x\\ | ||| 2 |
||||1 + cot |-|| | ||| / 2/x\\ |
|||\ \2// | ||| |1 + cot |-|| |
\\\ / \\\ \ \2// /
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/x pi\| |
|| 2 | ||| cos |- - --|| |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x pi\\ |
|| 2/x pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/x\ | |
|| 2 | ||| sec |-| | |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi x\| |
|| 2 | ||| csc |-- - -|| |
16 - 6*|
$$\left(- 6 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(6 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 16$$
16 - 6*Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True)) - 6*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True))