Inverse Trigonometric Functions

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is :

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^2+b^2$ is equal to

Given that the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[-1,1]$ such that $\cos ^{-1} x-\sin ^{-1} y=\alpha, \frac{-\pi}{2} \leq \alpha \leq \pi$. Then, the minimum value of $x^2+y^2+2 x y \sin \alpha$ is

${\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,$ then sin x is equal to :

If ${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,$ then $4{x^2} - 4xy\cos \alpha + {y^2}$ is equal to :

If sin⁻¹$\left( {{x \over 5}} \right)$ + cosec⁻¹$\left( {{5 \over 4}} \right)$ = ${\pi \over 2}$, then the value of x is :

The value of $cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)$ is :

If $x, y, z$ are in A.P. and ${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$ and ${\tan ^{ - 1}}z$ are also in A.P., then :

Let ${\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),$ where $\left| x \right| < {1 \over {\sqrt 3 }}.$ Then a value of $y$ is :

The value of tan⁻¹ $\left[ {{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} } \over {\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right],$ $\left| x \right| < {1 \over 2},x \ne 0,$ is equal to :