$\cot ^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right)=$ . . . . . . where,$x>1$.

If $0 < x < 1$,then $\sqrt{1 + x^2} [\{x \cos (\cot^{-1} x) + \sin (\cot^{-1} x)\} ^2 - 1]^{\frac{1}{2}} =$

$\cos \left(2 \cos ^{-1} \frac{1}{5}+\sin ^{-1} \frac{1}{5}\right)$ is equal to

If $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c}$ and $0 < x < 1,$ then the value of $\cos \left(\frac{\pi c}{a + b}\right)$ is

If $\sin ^{-1} a=\alpha+\beta$ and $\sin ^{-1} b=\alpha-\beta$ then,$\sin ^2 \alpha+\cos ^2 \beta=$ . . . . . . .

If $y = \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right]$,then $\frac{dy}{dx} = $