The number of real solutions of $\operatorname{Tan}^{-1} x + \operatorname{Tan}^{-1} 2x = \frac{\pi}{4}$ is

If $u = \tan^{-1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)$ and $v = 2 \tan^{-1} x$,then $\frac{du}{dv}$ is equal to

The range of the real valued function $f(x) = \operatorname{Cos}^{-1}(-x) + \operatorname{Sin}^{-1}(-x) + \operatorname{Cosec}^{-1}(x)$ is

For $k \in R$,let the solutions of the equation $\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k$,where $0 < |x| < \frac{1}{\sqrt{2}}$,be $\alpha$ and $\beta$,where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^{2}- bx -5=0$ are $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}$ and $\frac{\alpha}{\beta}$,then $\frac{b}{k^{2}}$ is equal to $......$

The imaginary part of $\tan^{-1}\left(\frac{5i}{3}\right)$ is

$\sec ^2(\tan ^{-1} 2) + \operatorname{cosec}^2(\cot ^{-1} 3)$ is equal to