Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

$\begin{array}{rl}& \text{ Let }{\tan }^{-1}\alpha =A\Rightarrow \tan A=\alpha \\ & {\cot }^{-1}\beta =B\Rightarrow \cot B=\beta \\ & {\sec }^{2}A+{\mathrm{cosec}}^{2}B=36\\ & \Rightarrow 1+{\tan }^{2}A+1+{\cot }^{2}B=36\\ & \Rightarrow {\alpha }^2+{\beta }^2=34\end{array}$ Also $\alpha +\beta =8$ (Given) $\begin{array}{rl}& \therefore (\alpha +\beta {)}^2=34+2\alpha \beta =64\\ & \Rightarrow \alpha \beta =15\end{array}$ $\Rightarrow \alpha ,\beta$ are roots of equation $\begin{array}{rl}& x^2-8x+15=0\\ & \Rightarrow (x-3)(x-5)=0\\ & \Rightarrow x=3,5\\ & \therefore \alpha =3,\beta =5(\alpha <\beta )\\ & \therefore {\alpha }^2+\beta =9+5=14\end{array}$