Inverse Trigono...

The value of 
$$\displaystyle sin^{-1} \left \{ (sin\,\pi/3) \dfrac{x}{\sqrt{x^2 + k^2 - kx}} \right \} - cos^{-1} \left \{ cos\pi /6 \,\dfrac{x}{\sqrt{x^2 + k^2 - kx}} \right \} $$
 $$ \left (where \,\dfrac{k}{2} < x < 2k , \,k > 0 \right ) $$ is 

$$cos^{-1}(cos(\dfrac{5\pi}{4}))$$ is given by

The value $$2tan^{-1}\Bigg[\sqrt{\dfrac{a-b}{a+b}}\tan\dfrac{\theta}{2}\Bigg]$$ is equal to

If  f(x)=$$sin^{-1}\Bigg(\dfrac{\sqrt{3}}{2}x-\dfrac{1}{2}\sqrt{1-x^2}\Bigg),-\dfrac{1}{2}\leq x\leq1$$, then f(x) is equal to

If $$sin^{-1}\dfrac{5}{x}+sin^{-1}\dfrac{12}{x}=\dfrac{\pi}{2}$$, then x is equal to

If  $$3\sin^{-1}\bigg(\dfrac{2x}{1+x^2}\bigg)-4\cos^{-1}\bigg(\dfrac{1-x^2}{1+x^2}\bigg)+2tan^{-1}\bigg(\dfrac{2x}{1-x^2}\bigg)=\dfrac{\pi}{3}$$, where $$|x|<1$$. then x is equal to

If $$x\in\bigg(\dfrac{-\pi}{2},\dfrac{\pi}{2}\bigg)$$, then the value $$\tan^{-1}\Bigg(\dfrac{\tan x}{4}\Bigg)+\tan^{-1}\Bigg(\dfrac{3\sin 2x}{5+3\cos 2x}\Bigg)$$  is

The value of $$sin^{-1}\Bigg(cot\bigg(sin^{-1}\sqrt{\dfrac{2-\sqrt{3}}{4}}+cos^{-1}\dfrac{\sqrt{12}}{4}+\sec^{-1}\sqrt{2}\bigg)\Bigg)$$ is

If $$\tan^{-1}\dfrac{1-x}{1+x}=\dfrac{1}{2}\tan^{-1}x$$, then x is equal to

If  $$tan^{-1}x+2 cot^{-1}x=\dfrac{2\pi}{3}$$, then x is equal to