Inverse Trigonometric Functions

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