### Inverse Trigonometric Functions

goals
Inverse circular functions,Principal values of ${ sin }^{ -1 }x,{ cos }^{ -1 }x,{ tan }^{ -1 }x$.
${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\frac { x+y }{ 1-xy }$,      $xy<1$
$\pi +{ tan }^{ -1 }\frac { x+y }{ 1-xy }$,         $xy>1$.
Prove that
${ tan }^{ -1 }\left( \frac { 1 }{ 2 } tan2A \right) +{ tan }^{ -1 }(cotA)+{ tan }^{ -1 }({ cot }^{ 3 }A)$
$0$ if $\pi /4<A<\pi /2$
and $=\pi$ if $0<A<\pi /4$.
Prove that :
$\tan^{-1}{\dfrac{1}{2}}+\tan^{-1}{2}=\dfrac{\pi}{2}$
$\displaystyle \alpha = tan^{-1} (\frac{1}{2}) + tan^{-1} (\frac{1}{3}), \beta = cos^{-1} (\frac{2}{3}) + cos^{-1} (\frac{\sqrt{5}}{3})$ and $\gamma = sin^{-1} (sin (\displaystyle \frac{2\pi}{3})) + \frac{1}{2} cos^{-1} (cos (\frac{2\pi}{3}))$
If $y= \tan^{-1}[\dfrac{ a\cos x+ b \sin x}{ b\cos x- a \sin x}]$, find$\dfrac{dy}{dx}$.
If $\sin^{-1}\left(x-\dfrac {x^{2}}{2}+\dfrac {x^{3}}{4}-...\infty \right)+\cos^{-1}\left(x^{2}-\dfrac {x^{4}}{2}+\dfrac {x^{6}}{4}-...\infty \right)=\dfrac {\pi}{2}$ for $0 < |x| < \sqrt {2}$,then $x$ equal