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