Господин Экзамен
Общий знаменатель sqrt(1-sin(x)/(1+sin(x)))+sqrt(1+sin(x)/(1-sin(x)))
Решение
Вы ввели
[src]
________________ ________________ / sin(x) / sin(x) / 1 - ---------- + / 1 + ---------- \/ 1 + sin(x) \/ 1 - sin(x)
$$\sqrt{1 + \frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + 1}} + \sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}}$$
sqrt(1 - sin(x)/(1 + sin(x))) + sqrt(1 + sin(x)/(1 - sin(x)))
Общее упрощение
[src]
____________ _____________ / 1 / -1 / ---------- + / ----------- \/ 1 + sin(x) \/ -1 + sin(x)
$$\sqrt{- \frac{1}{\sin{\left(x \right)} - 1}} + \sqrt{\frac{1}{\sin{\left(x \right)} + 1}}$$
sqrt(1/(1 + sin(x))) + sqrt(-1/(-1 + sin(x)))
Собрать выражение
[src]
________________ _________________ / sin(x) / sin(x) / 1 - ---------- + / 1 - ----------- \/ 1 + sin(x) \/ -1 + sin(x)
$$\sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - 1}} + \sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}}$$
sqrt(1 - sin(x)/(1 + sin(x))) + sqrt(1 - sin(x)/(-1 + sin(x)))
Тригонометрическая часть
[src]
____________ _____________ / 1 / -1 / ---------- + / ----------- \/ 1 + sin(x) \/ -1 + sin(x)
$$\sqrt{- \frac{1}{\sin{\left(x \right)} - 1}} + \sqrt{\frac{1}{\sin{\left(x \right)} + 1}}$$
____________ _____________
/ 1 / -1
/ ---------- + / -----------
/ 1 / 1
/ 1 + ------ / -1 + ------
\/ csc(x) \/ csc(x)
$$\sqrt{- \frac{1}{-1 + \frac{1}{\csc{\left(x \right)}}}} + \sqrt{\frac{1}{1 + \frac{1}{\csc{\left(x \right)}}}}$$
_________________ __________________
/ 1 / -1
/ --------------- + / ----------------
/ / pi\ / / pi\
/ 1 + cos|x - --| / -1 + cos|x - --|
\/ \ 2 / \/ \ 2 /
$$\sqrt{- \frac{1}{\cos{\left(x - \frac{\pi}{2} \right)} - 1}} + \sqrt{\frac{1}{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}$$
_________________ __________________
/ 1 / -1
/ --------------- + / ----------------
/ 1 / 1
/ 1 + ----------- / -1 + -----------
/ / pi\ / / pi\
/ sec|x - --| / sec|x - --|
\/ \ 2 / \/ \ 2 /
$$\sqrt{- \frac{1}{-1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}} + \sqrt{\frac{1}{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}}$$
_________________________ _________________________
/ 1 / 1
/ 1 + ------------------- + / 1 - -------------------
/ / 1 \ / / 1 \
/ |1 - ------|*csc(x) / |1 + ------|*csc(x)
\/ \ csc(x)/ \/ \ csc(x)/
$$\sqrt{1 + \frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}} + \sqrt{1 - \frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}}$$
_____________________ _____________________
/ / pi\ / / pi\
/ cos|x - --| / cos|x - --|
/ \ 2 / / \ 2 /
/ 1 + --------------- + / 1 - ---------------
/ / pi\ / / pi\
/ 1 - cos|x - --| / 1 + cos|x - --|
\/ \ 2 / \/ \ 2 /
$$\sqrt{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{- \cos{\left(x - \frac{\pi}{2} \right)} + 1}} + \sqrt{1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}$$
_________________ __________________
/ 1 / -1
/ --------------- + / ----------------
/ /x\ / /x\
/ 2*tan|-| / 2*tan|-|
/ \2/ / \2/
/ 1 + ----------- / -1 + -----------
/ 2/x\ / 2/x\
/ 1 + tan |-| / 1 + tan |-|
\/ \2/ \/ \2/
$$\sqrt{- \frac{1}{-1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}} + \sqrt{\frac{1}{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}}$$
___________________________________ ___________________________________
/ 1 / 1
/ 1 + ----------------------------- + / 1 - -----------------------------
/ / 1 \ / / 1 \
/ |1 - -----------|*csc(pi - x) / |1 + -----------|*csc(pi - x)
\/ \ csc(pi - x)/ \/ \ csc(pi - x)/
$$\sqrt{1 + \frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \pi \right)}}\right) \csc{\left(- x + \pi \right)}}} + \sqrt{1 - \frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \pi \right)}}\right) \csc{\left(- x + \pi \right)}}}$$
____________________________ _____________________________
/ 1 / -1
/ -------------------------- + / ---------------------------
/ /x\ /x\ / /x\ /x\
/ 1 + cos(x)*tan|-| + tan|-| / -1 + cos(x)*tan|-| + tan|-|
\/ \2/ \2/ \/ \2/ \2/
$$\sqrt{- \frac{1}{\cos{\left(x \right)} \tan{\left(\frac{x}{2} \right)} + \tan{\left(\frac{x}{2} \right)} - 1}} + \sqrt{\frac{1}{\cos{\left(x \right)} \tan{\left(\frac{x}{2} \right)} + \tan{\left(\frac{x}{2} \right)} + 1}}$$
___________________________________ ___________________________________
/ 1 / 1
/ 1 + ----------------------------- + / 1 - -----------------------------
/ / 1 \ / pi\ / / 1 \ / pi\
/ |1 - -----------|*sec|x - --| / |1 + -----------|*sec|x - --|
/ | / pi\| \ 2 / / | / pi\| \ 2 /
/ | sec|x - --|| / | sec|x - --||
\/ \ \ 2 // \/ \ \ 2 //
$$\sqrt{1 + \frac{1}{\left(1 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}} + \sqrt{1 - \frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}}$$
___________________________________ ___________________________________
/ 1 / 1
/ 1 + ----------------------------- + / 1 - -----------------------------
/ / 1 \ /pi \ / / 1 \ /pi \
/ |1 - -----------|*sec|-- - x| / |1 + -----------|*sec|-- - x|
/ | /pi \| \2 / / | /pi \| \2 /
/ | sec|-- - x|| / | sec|-- - x||
\/ \ \2 // \/ \ \2 //
$$\sqrt{1 + \frac{1}{\left(1 - \frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}}\right) \sec{\left(- x + \frac{\pi}{2} \right)}}} + \sqrt{1 - \frac{1}{\left(1 + \frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}}\right) \sec{\left(- x + \frac{\pi}{2} \right)}}}$$
_________________________________ __________________________________
/ 1 / -1
/ ------------------------------- + / --------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ 1 + |< | / -1 + |< |
\/ \\sin(x) otherwise / \/ \\sin(x) otherwise /
$$\left(\sqrt{- \frac{1}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) - 1}}\right) + \left(\sqrt{\frac{1}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) + 1}}\right)$$
_____________________________________________________
/ 2/x\
/ 8*sin |-|*(1 - cos(x))*sin(x) ___________________________
/ \2/ ___ / 1 - cos(x)
/ 1 + ----------------------------------------------- + \/ 2 * / -------------------------
/ 2 / 2
/ / ___ / pi\\ / 2 4/x\\ / / ___ / pi\\
/ |-1 + \/ 2 *sin|x + --|| *|sin (x) + 4*sin |-|| / |-1 + \/ 2 *cos|x + --||
\/ \ \ 4 // \ \2// \/ \ \ 4 //
$$\sqrt{2} \sqrt{\frac{- \cos{\left(x \right)} + 1}{\left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}} + \sqrt{\frac{8 \cdot \left(- \cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2} \cdot \left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)} + 1}$$
_____________________________________ _____________________________________
/ /x\ / /x\
/ 2*tan|-| / 2*tan|-|
/ \2/ / \2/
/ 1 - ------------------------------- + / 1 + -------------------------------
/ / /x\ \ / / /x\ \
/ | 2*tan|-| | / | 2*tan|-| |
/ / 2/x\\ | \2/ | / / 2/x\\ | \2/ |
/ |1 + tan |-||*|1 + -----------| / |1 + tan |-||*|1 - -----------|
/ \ \2// | 2/x\| / \ \2// | 2/x\|
/ | 1 + tan |-|| / | 1 + tan |-||
\/ \ \2// \/ \ \2//
$$\sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}} + \sqrt{1 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}}$$
_____________________________________ _____________________________________
/ /x\ / /x\
/ 2*cot|-| / 2*cot|-|
/ \2/ / \2/
/ 1 - ------------------------------- + / 1 + -------------------------------
/ / /x\ \ / / /x\ \
/ | 2*cot|-| | / | 2*cot|-| |
/ / 2/x\\ | \2/ | / / 2/x\\ | \2/ |
/ |1 + cot |-||*|1 + -----------| / |1 + cot |-||*|1 - -----------|
/ \ \2// | 2/x\| / \ \2// | 2/x\|
/ | 1 + cot |-|| / | 1 + cot |-||
\/ \ \2// \/ \ \2//
$$\sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}} + \sqrt{1 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}}$$
______________________________________ _______________________________________
/ 1 / -1
/ ------------------------------------ + / -------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || /x\ | / || /x\ |
/ || 2*cot|-| | / || 2*cot|-| |
/ 1 + |< \2/ | / -1 + |< \2/ |
/ ||----------- otherwise | / ||----------- otherwise |
/ || 2/x\ | / || 2/x\ |
/ ||1 + cot |-| | / ||1 + cot |-| |
\/ \\ \2/ / \/ \\ \2/ /
$$\left(\sqrt{- \frac{1}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1}}\right) + \left(\sqrt{\frac{1}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}}\right)$$
_____________________________________________________ _____________________________________________________
/ 2 / 2
/ 1 - ----------------------------------------------- + / 1 + -----------------------------------------------
/ / 1 \ / 2 \ /x\ / / 1 \ / 2 \ /x\
/ |1 + -------|*|1 + --------------------|*cot|-| / |1 + -------|*|1 - --------------------|*cot|-|
/ | 2/x\| | / 1 \ /x\| \2/ / | 2/x\| | / 1 \ /x\| \2/
/ | cot |-|| | |1 + -------|*cot|-|| / | cot |-|| | |1 + -------|*cot|-||
/ \ \2// | | 2/x\| \2/| / \ \2// | | 2/x\| \2/|
/ | | cot |-|| | / | | cot |-|| |
\/ \ \ \2// / \/ \ \ \2// /
$$\sqrt{1 + \frac{2}{\left(1 - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}} + \sqrt{1 - \frac{2}{\left(1 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}}$$
_____________________________________ _____________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ < / <
/ \sin(x) otherwise / \sin(x) otherwise
/ 1 + ------------------------------- + / 1 - -------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ 1 - |< | / 1 + |< |
\/ \\sin(x) otherwise / \/ \\sin(x) otherwise /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_____________________________________________ _____________________________________________
/ / 2/x pi\\ / / 2/x pi\\
/ |1 - cot |- + --||*(1 + sin(x)) / |1 - cot |- + --||*(1 + sin(x))
/ \ \2 4 // / \ \2 4 //
/ 1 + --------------------------------------- + / 1 - ---------------------------------------
/ / / 2/x pi\\ \ / / / 2/x pi\\ \
/ | |1 - cot |- + --||*(1 + sin(x))| / | |1 - cot |- + --||*(1 + sin(x))|
/ | \ \2 4 // | / | \ \2 4 // |
/ 2*|1 - -------------------------------| / 2*|1 + -------------------------------|
\/ \ 2 / \/ \ 2 /
$$\sqrt{\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right)}{2 \left(- \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right)}{2} + 1\right)} + 1} + \sqrt{- \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right)}{2 \left(\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right)}{2} + 1\right)} + 1}$$
________________________________________________ ________________________________________________
/ 2/x pi\ / 2/x pi\
/ -1 + tan |- + --| / -1 + tan |- + --|
/ \2 4 / / \2 4 /
/ 1 + ------------------------------------------ + / 1 - ------------------------------------------
/ / 2/x pi\\ / / 2/x pi\\
/ | -1 + tan |- + --|| / | -1 + tan |- + --||
/ / 2/x pi\\ | \2 4 /| / / 2/x pi\\ | \2 4 /|
/ |1 + tan |- + --||*|1 - -----------------| / |1 + tan |- + --||*|1 + -----------------|
/ \ \2 4 // | 2/x pi\| / \ \2 4 // | 2/x pi\|
/ | 1 + tan |- + --|| / | 1 + tan |- + --||
\/ \ \2 4 // \/ \ \2 4 //
$$\sqrt{1 + \frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\left(- \frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}} + \sqrt{1 - \frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\left(\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}}$$
_______________________________________________ _______________________________________________
/ 2/x pi\ / 2/x pi\
/ 1 - cot |- + --| / 1 - cot |- + --|
/ \2 4 / / \2 4 /
/ 1 + ----------------------------------------- + / 1 - -----------------------------------------
/ / 2/x pi\\ / / 2/x pi\\
/ | 1 - cot |- + --|| / | 1 - cot |- + --||
/ / 2/x pi\\ | \2 4 /| / / 2/x pi\\ | \2 4 /|
/ |1 + cot |- + --||*|1 - ----------------| / |1 + cot |- + --||*|1 + ----------------|
/ \ \2 4 // | 2/x pi\| / \ \2 4 // | 2/x pi\|
/ | 1 + cot |- + --|| / | 1 + cot |- + --||
\/ \ \2 4 // \/ \ \2 4 //
$$\sqrt{1 + \frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{\left(- \frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}} + \sqrt{1 - \frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{\left(\frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}}$$
_____________________________________ _____________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ < 1 / < 1
/ |------ otherwise / |------ otherwise
/ \csc(x) / \csc(x)
/ 1 + ------------------------------- + / 1 - -------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ 1 - |< 1 | / 1 + |< 1 |
/ ||------ otherwise | / ||------ otherwise |
\/ \\csc(x) / \/ \\csc(x) /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
__________________________________________ __________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ < / pi\ / < / pi\
/ |cos|x - --| otherwise / |cos|x - --| otherwise
/ \ \ 2 / / \ \ 2 /
/ 1 + ------------------------------------ + / 1 - ------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ 1 - |< / pi\ | / 1 + |< / pi\ |
/ ||cos|x - --| otherwise | / ||cos|x - --| otherwise |
\/ \\ \ 2 / / \/ \\ \ 2 / /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
__________________________________________ __________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | 1 / | 1
/ <----------- otherwise / <----------- otherwise
/ | / pi\ / | / pi\
/ |sec|x - --| / |sec|x - --|
/ \ \ 2 / / \ \ 2 /
/ 1 + ------------------------------------ + / 1 - ------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || 1 | / || 1 |
/ 1 - |<----------- otherwise | / 1 + |<----------- otherwise |
/ || / pi\ | / || / pi\ |
/ ||sec|x - --| | / ||sec|x - --| |
\/ \\ \ 2 / / \/ \\ \ 2 / /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
________________________________________________ ________________________________________________
/ / / 3*pi\ / / / 3*pi\
/ | 1 for |x + ----| mod 2*pi = 0 / | 1 for |x + ----| mod 2*pi = 0
/ < \ 2 / / < \ 2 /
/ | / |
/ \sin(x) otherwise / \sin(x) otherwise
/ 1 + ------------------------------------------ + / 1 - ------------------------------------------
/ // / 3*pi\ \ / // / 3*pi\ \
/ || 1 for |x + ----| mod 2*pi = 0| / || 1 for |x + ----| mod 2*pi = 0|
/ 1 - |< \ 2 / | / 1 + |< \ 2 / |
/ || | / || |
\/ \\sin(x) otherwise / \/ \\sin(x) otherwise /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_________________________________________________________ _________________________________________________________
/ 2/x\ / 2/x\
/ 4*sin |-| / 4*sin |-|
/ \2/ / \2/
/ 1 - --------------------------------------------------- + / 1 + ---------------------------------------------------
/ / 4/x\\ / 2/x\ \ / / 4/x\\ / 2/x\ \
/ | 4*sin |-|| | 4*sin |-| | / | 4*sin |-|| | 4*sin |-| |
/ | \2/| | \2/ | / | \2/| | \2/ |
/ |1 + ---------|*|1 + ----------------------|*sin(x) / |1 + ---------|*|1 - ----------------------|*sin(x)
/ | 2 | | / 4/x\\ | / | 2 | | / 4/x\\ |
/ \ sin (x) / | | 4*sin |-|| | / \ sin (x) / | | 4*sin |-|| |
/ | | \2/| | / | | \2/| |
/ | |1 + ---------|*sin(x)| / | |1 + ---------|*sin(x)|
/ | | 2 | | / | | 2 | |
\/ \ \ sin (x) / / \/ \ \ sin (x) / /
$$\sqrt{1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}} + \sqrt{1 - \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}}$$
_________________________________________ _________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ |1 - cos(x) / |1 - cos(x)
/ <---------- otherwise / <---------- otherwise
/ | /x\ / | /x\
/ | tan|-| / | tan|-|
/ \ \2/ / \ \2/
/ 1 + ----------------------------------- + / 1 - -----------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ ||1 - cos(x) | / ||1 - cos(x) |
/ 1 - |<---------- otherwise | / 1 + |<---------- otherwise |
/ || /x\ | / || /x\ |
/ || tan|-| | / || tan|-| |
\/ \\ \2/ / \/ \\ \2/ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
__________________________________________ __________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | /x\ / | /x\
/ | 2*tan|-| / | 2*tan|-|
/ < \2/ / < \2/
/ |----------- otherwise / |----------- otherwise
/ | 2/x\ / | 2/x\
/ |1 + tan |-| / |1 + tan |-|
/ \ \2/ / \ \2/
/ 1 + ------------------------------------ + / 1 - ------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || /x\ | / || /x\ |
/ || 2*tan|-| | / || 2*tan|-| |
/ 1 - |< \2/ | / 1 + |< \2/ |
/ ||----------- otherwise | / ||----------- otherwise |
/ || 2/x\ | / || 2/x\ |
/ ||1 + tan |-| | / ||1 + tan |-| |
\/ \\ \2/ / \/ \\ \2/ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
__________________________________________ __________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | /x\ / | /x\
/ | 2*cot|-| / | 2*cot|-|
/ < \2/ / < \2/
/ |----------- otherwise / |----------- otherwise
/ | 2/x\ / | 2/x\
/ |1 + cot |-| / |1 + cot |-|
/ \ \2/ / \ \2/
/ 1 + ------------------------------------ + / 1 - ------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || /x\ | / || /x\ |
/ || 2*cot|-| | / || 2*cot|-| |
/ 1 - |< \2/ | / 1 + |< \2/ |
/ ||----------- otherwise | / ||----------- otherwise |
/ || 2/x\ | / || 2/x\ |
/ ||1 + cot |-| | / ||1 + cot |-| |
\/ \\ \2/ / \/ \\ \2/ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
________________________________________________________ ________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
___________________________________________________ ___________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | 2 / | 2
/ |-------------------- otherwise / |-------------------- otherwise
/
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_______________________________________________________________ _______________________________________________________________
/ /x pi\ / /x pi\
/ 2*cos|- - --| / 2*cos|- - --|
/ \2 2 / / \2 2 /
/ 1 - --------------------------------------------------------- + / 1 + ---------------------------------------------------------
/ / 2/x pi\\ / /x pi\ \ / / 2/x pi\\ / /x pi\ \
/ | cos |- - --|| | 2*cos|- - --| | / | cos |- - --|| | 2*cos|- - --| |
/ | \2 2 /| | \2 2 / | /x\ / | \2 2 /| | \2 2 / | /x\
/ |1 + ------------|*|1 + -------------------------|*cos|-| / |1 + ------------|*|1 - -------------------------|*cos|-|
/ | 2/x\ | | / 2/x pi\\ | \2/ / | 2/x\ | | / 2/x pi\\ | \2/
/ | cos |-| | | | cos |- - --|| | / | cos |-| | | | cos |- - --|| |
/ \ \2/ / | | \2 2 /| /x\| / \ \2/ / | | \2 2 /| /x\|
/ | |1 + ------------|*cos|-|| / | |1 + ------------|*cos|-||
/ | | 2/x\ | \2/| / | | 2/x\ | \2/|
/ | | cos |-| | | / | | cos |-| | |
\/ \ \ \2/ / / \/ \ \ \2/ / /
$$\sqrt{1 + \frac{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}} + \sqrt{1 - \frac{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}}$$
_________________________________________________________________________ _________________________________________________________________________
/ /x\ / /x\
/ 2*sec|-| / 2*sec|-|
/ \2/ / \2/
/ 1 - ------------------------------------------------------------------- + / 1 + -------------------------------------------------------------------
/ / 2/x\ \ / /x\ \ / / 2/x\ \ / /x\ \
/ | sec |-| | | 2*sec|-| | / | sec |-| | | 2*sec|-| |
/ | \2/ | | \2/ | /x pi\ / | \2/ | | \2/ | /x pi\
/ |1 + ------------|*|1 + ------------------------------|*sec|- - --| / |1 + ------------|*|1 - ------------------------------|*sec|- - --|
/ | 2/x pi\| | / 2/x\ \ | \2 2 / / | 2/x pi\| | / 2/x\ \ | \2 2 /
/ | sec |- - --|| | | sec |-| | | / | sec |- - --|| | | sec |-| | |
/ \ \2 2 // | | \2/ | /x pi\| / \ \2 2 // | | \2/ | /x pi\|
/ | |1 + ------------|*sec|- - --|| / | |1 + ------------|*sec|- - --||
/ | | 2/x pi\| \2 2 /| / | | 2/x pi\| \2 2 /|
/ | | sec |- - --|| | / | | sec |- - --|| |
\/ \ \ \2 2 // / \/ \ \ \2 2 // /
$$\sqrt{1 + \frac{2 \sec{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}} + \sqrt{1 - \frac{2 \sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}$$
_______________________________________________________________ _______________________________________________________________
/ /pi x\ / /pi x\
/ 2*csc|-- - -| / 2*csc|-- - -|
/ \2 2/ / \2 2/
/ 1 - --------------------------------------------------------- + / 1 + ---------------------------------------------------------
/ / 2/pi x\\ / /pi x\ \ / / 2/pi x\\ / /pi x\ \
/ | csc |-- - -|| | 2*csc|-- - -| | / | csc |-- - -|| | 2*csc|-- - -| |
/ | \2 2/| | \2 2/ | /x\ / | \2 2/| | \2 2/ | /x\
/ |1 + ------------|*|1 + -------------------------|*csc|-| / |1 + ------------|*|1 - -------------------------|*csc|-|
/ | 2/x\ | | / 2/pi x\\ | \2/ / | 2/x\ | | / 2/pi x\\ | \2/
/ | csc |-| | | | csc |-- - -|| | / | csc |-| | | | csc |-- - -|| |
/ \ \2/ / | | \2 2/| /x\| / \ \2/ / | | \2 2/| /x\|
/ | |1 + ------------|*csc|-|| / | |1 + ------------|*csc|-||
/ | | 2/x\ | \2/| / | | 2/x\ | \2/|
/ | | csc |-| | | / | | csc |-| | |
\/ \ \ \2/ / / \/ \ \ \2/ / /
$$\sqrt{1 + \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}} + \sqrt{1 - \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}}$$
___________________________________________________________ ___________________________________________________________
/ / / 3*pi\ / / / 3*pi\
/ | 1 for |x + ----| mod 2*pi = 0 / | 1 for |x + ----| mod 2*pi = 0
/ | \ 2 / / | \ 2 /
/ | / |
/ | 2/x pi\ / | 2/x pi\
/ <-1 + tan |- + --| / <-1 + tan |- + --|
/ | \2 4 / / | \2 4 /
/ |----------------- otherwise / |----------------- otherwise
/ | 2/x pi\ / | 2/x pi\
/ | 1 + tan |- + --| / | 1 + tan |- + --|
/ \ \2 4 / / \ \2 4 /
/ 1 + ----------------------------------------------------- + / 1 - -----------------------------------------------------
/ // / 3*pi\ \ / // / 3*pi\ \
/ || 1 for |x + ----| mod 2*pi = 0| / || 1 for |x + ----| mod 2*pi = 0|
/ || \ 2 / | / || \ 2 / |
/ || | / || |
/ || 2/x pi\ | / || 2/x pi\ |
/ 1 - |<-1 + tan |- + --| | / 1 + |<-1 + tan |- + --| |
/ || \2 4 / | / || \2 4 / |
/ ||----------------- otherwise | / ||----------------- otherwise |
/ || 2/x pi\ | / || 2/x pi\ |
/ || 1 + tan |- + --| | / || 1 + tan |- + --| |
\/ \\ \2 4 / / \/ \\ \2 4 / /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
______________________________________________________ ______________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | sin(x) / | sin(x)
/ |----------------------- otherwise / |----------------------- otherwise
/ |/ 2 \ / |/ 2 \
/ <| sin (x) | 2/x\ / <| sin (x) | 2/x\
/ ||1 + ---------|*sin |-| / ||1 + ---------|*sin |-|
/ || 4/x\| \2/ / || 4/x\| \2/
/ || 4*sin |-|| / || 4*sin |-||
/ |\ \2// / |\ \2//
/ \ / \
/ 1 + ------------------------------------------------ + / 1 - ------------------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || sin(x) | / || sin(x) |
/ ||----------------------- otherwise | / ||----------------------- otherwise |
/ ||/ 2 \ | / ||/ 2 \ |
/ 1 - |<| sin (x) | 2/x\ | / 1 + |<| sin (x) | 2/x\ |
/ |||1 + ---------|*sin |-| | / |||1 + ---------|*sin |-| |
/ ||| 4/x\| \2/ | / ||| 4/x\| \2/ |
/ ||| 4*sin |-|| | / ||| 4*sin |-|| |
/ ||\ \2// | / ||\ \2// |
\/ \\ / \/ \\ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_____________________________________________________________ _____________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ |/ 0 for x mod pi = 0 / |/ 0 for x mod pi = 0
/ || / ||
/ || /x\ / || /x\
/ <| 2*cot|-| / <| 2*cot|-|
/ |< \2/ otherwise / |< \2/ otherwise
/ ||----------- otherwise / ||----------- otherwise
/ || 2/x\ / || 2/x\
/ ||1 + cot |-| / ||1 + cot |-|
/ \\ \2/ / \\ \2/
/ 1 + ------------------------------------------------------- + / 1 - -------------------------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ ||/ 0 for x mod pi = 0 | / ||/ 0 for x mod pi = 0 |
/ ||| | / ||| |
/ ||| /x\ | / ||| /x\ |
/ 1 - |<| 2*cot|-| | / 1 + |<| 2*cot|-| |
/ ||< \2/ otherwise | / ||< \2/ otherwise |
/ |||----------- otherwise | / |||----------- otherwise |
/ ||| 2/x\ | / ||| 2/x\ |
/ |||1 + cot |-| | / |||1 + cot |-| |
\/ \\\ \2/ / \/ \\\ \2/ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
___________________________________________________________ ___________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | 2*sin(x) / | 2*sin(x)
/ |---------------------------- otherwise / |---------------------------- otherwise
/ | / 2 \ / | / 2 \
/ < | sin (x) | / < | sin (x) |
/ |(1 - cos(x))*|1 + ---------| / |(1 - cos(x))*|1 + ---------|
/ | | 4/x\| / | | 4/x\|
/ | | 4*sin |-|| / | | 4*sin |-||
/ | \ \2// / | \ \2//
/ \ / \
/ 1 + ----------------------------------------------------- + / 1 - -----------------------------------------------------
/ // 0 for x mod pi = 0\ / // 0 for x mod pi = 0\
/ || | / || |
/ || 2*sin(x) | / || 2*sin(x) |
/ ||---------------------------- otherwise | / ||---------------------------- otherwise |
/ || / 2 \ | / || / 2 \ |
/ 1 - |< | sin (x) | | / 1 + |< | sin (x) | |
/ ||(1 - cos(x))*|1 + ---------| | / ||(1 - cos(x))*|1 + ---------| |
/ || | 4/x\| | / || | 4/x\| |
/ || | 4*sin |-|| | / || | 4*sin |-|| |
/ || \ \2// | / || \ \2// |
\/ \\ / \/ \\ /
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
________________________________________________________ ________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | /x pi\ / | /x pi\
/ | 2*sec|- - --| / | 2*sec|- - --|
/ | \2 2 / / | \2 2 /
/ |------------------------- otherwise / |------------------------- otherwise
/
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_____________________________________________________________ _____________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | /x\ / | /x\
/ | 2*cos|-| / | 2*cos|-|
/ | \2/ / | \2/
/ |------------------------------ otherwise / |------------------------------ otherwise
/
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
_____________________________________________________________ _____________________________________________________________
/ / 0 for x mod pi = 0 / / 0 for x mod pi = 0
/ | / |
/ | /x\ / | /x\
/ | 2*csc|-| / | 2*csc|-|
/ | \2/ / | \2/
/ |------------------------------ otherwise / |------------------------------ otherwise
/
$$\left(\sqrt{1 + \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right) + \left(\sqrt{1 - \left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)}\right)$$
sqrt(1 + Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))/(1 - Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True)))) + sqrt(1 - Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))/(1 + Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))))
Рациональный знаменатель
[src]
________________ _________________ / sin(x) / sin(x) / 1 - ---------- + / 1 - ----------- \/ 1 + sin(x) \/ -1 + sin(x)
$$\sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - 1}} + \sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}}$$
sqrt(1 - sin(x)/(1 + sin(x))) + sqrt(1 - sin(x)/(-1 + sin(x)))
Комбинаторика
[src]
____________ ____________ / 1 / 1 / ---------- + / ---------- \/ 1 - sin(x) \/ 1 + sin(x)
$$\sqrt{\frac{1}{- \sin{\left(x \right)} + 1}} + \sqrt{\frac{1}{\sin{\left(x \right)} + 1}}$$
sqrt(1/(1 - sin(x))) + sqrt(1/(1 + sin(x)))
Объединение рациональных выражений
[src]
____________ ____________ / 1 / 1 / ---------- + / ---------- \/ 1 - sin(x) \/ 1 + sin(x)
$$\sqrt{\frac{1}{- \sin{\left(x \right)} + 1}} + \sqrt{\frac{1}{\sin{\left(x \right)} + 1}}$$
sqrt(1/(1 - sin(x))) + sqrt(1/(1 + sin(x)))
Общий знаменатель
[src]
________________ _________________ / sin(x) / sin(x) / 1 - ---------- + / 1 - ----------- \/ 1 + sin(x) \/ -1 + sin(x)
$$\sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - 1}} + \sqrt{1 - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}}$$
sqrt(1 - sin(x)/(1 + sin(x))) + sqrt(1 - sin(x)/(-1 + sin(x)))
Степени
[src]
________________________________ ________________________________
/ / -I*x I*x\ / / -I*x I*x\
/ I*\- e + e / / I*\- e + e /
/ 1 + -------------------------- + / 1 - --------------------------
/ / / -I*x I*x\\ / / / -I*x I*x\\
/ | I*\- e + e /| / | I*\- e + e /|
/ 2*|1 - ------------------| / 2*|1 + ------------------|
\/ \ 2 / \/ \ 2 /
$$\sqrt{1 + \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)}} + \sqrt{1 - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)}}$$
sqrt(1 + i*(-exp(-i*x) + exp(i*x))/(2*(1 - i*(-exp(-i*x) + exp(i*x))/2))) + sqrt(1 - i*(-exp(-i*x) + exp(i*x))/(2*(1 + i*(-exp(-i*x) + exp(i*x))/2)))
Численный ответ
[src]
(1.0 + sin(x)/(1.0 - sin(x)))^0.5 + (1.0 - sin(x)/(1.0 + sin(x)))^0.5
(1.0 + sin(x)/(1.0 - sin(x)))^0.5 + (1.0 - sin(x)/(1.0 + sin(x)))^0.5