Find the value of ${\sec }^2\left({\tan }^{-1}2\right)+{cosec}^2\left({\cot }^{-1}3\right)+2$.

Find the value of $\tan \left[{\sin }^{-1}(3/5)+{\cot }^{-1}(3/2)\right]$.

Find the number of real solutions: ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\mathrm{\pi }}{2}$

Find the number of real solutions:

$\frac{\pi }{2}+{\cos }^{-1}\left(\cos x\right)=\left|\tan x\right|, 0\le x\le 2\pi$

Prove that:

${\tan }^{-1}\left(\frac{c_{1}x-y}{c_{1}y+x}\right)+{\tan }^{-1}\left(\frac{c_2-c_1}{1+c_2{c}_1}\right)+{\tan }^{-1}\left(\frac{c_3-c_2}{1+c_3{c}_2}\right)+\dots +\dots +{\tan }^{-1}\left(\frac{1}{c_n}\right)={\tan }^{-1}\left(\frac{x}{y}\right).$

Find the sum of the infinite series:

${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+{\sin }^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}\right)+\dots +{\sin }^{-1}\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n}(n+1)}\right)+\dots$

Find the sum of the $n$ terms of the series:

${\cot }^{-1}\left(2·1^2\right)+{\cot }^{-1}\left(2·2^2\right)+{\cot }^{-1}\left(2·3^2\right)+\dots +\dots$ and hence the sum of infinite series.

If $\frac{m\tan (\alpha -\theta )}{{\cos }^2\theta }=\frac{n\tan \theta }{{\cos }^2(\alpha -\theta )}$, then prove that $\theta =\frac{1}{2}\left[\alpha -{\tan }^{-1}\left(\left(\frac{n-m}{n+m}\right)\tan \alpha \right)\right]$.