If $\sin ^{-1} x + \sin ^{-1} y = \frac{\pi}{2}$,then $x^{2}$ is equal to

$\cos \left(2 \left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} 5\right)\right) = $ . . . . . .

If $\sec^{-1} \left( \frac{x^2 + y^2}{x^2 - y^2} \right) = 2a$, such that $y \frac{dy}{dx} = x \cdot f(a)$, then the value of $f \left( \frac{2\pi}{3} \right)$ is

Find the number of solutions for the equation $\sin^{-1} x = 2 \tan^{-1} x$ (in principal values).

For $n \in N$,if $\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}$,then $n$ is equal to .........

Consider the following statements.
$I$. $\sin ^{-1}(y^2-4y+6)+\cos ^{-1}(y^2-4y+6) = \frac{\pi}{2}, \forall y \in R$
$II$. $\sec ^{-1}(y^2-4y+6)+\operatorname{cosec}^{-1}(y^2-4y+6) = \frac{\pi}{2}, \forall y \in R$
Which of the above statement$(s)$ is/are true?