If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta-8$ and $\sec ^2\left( an ^{-1} \alpha ight)+\operatorname{cosec}^2\left(\cot ^{-1} \beta ight)-36$, then $\alpha^2+\beta$ is_______.

Question

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta-8$ and $\sec ^2\left(	an ^{-1} \alphaight)+\operatorname{cosec}^2\left(\cot ^{-1} \betaight)-36$, then $\alpha^2+\beta$ is_______.

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Let } 	an ^{-1} \alpha=A \Rightarrow 	an A=\alpha \ & \cot ^{-1} \beta=B \Rightarrow \cot B=\beta \ & \sec ^2 A+\operatorname{cosec}^2 B=36 \ & \Rightarrow 1+	an ^2 A+1+\cot ^2 B=36 \ & \Rightarrow \alpha^2+\beta^2=34 \end{aligned}$ Also $\alpha+\beta=8$ (Given) $\begin{aligned} & 	herefore(\alpha+\beta)^2=34+2 \alpha \beta=64 \ & \Rightarrow \alpha \beta=15 \end{aligned}$ $\Rightarrow \alpha, \beta$ are roots of equation $\begin{aligned} & x^2-8 x+15=0 \ & \Rightarrow(x-3)(x-5)=0 \ & \Rightarrow \quad x=3,5 \ & 	herefore \quad \alpha=3, \beta=5 \quad(\alpha \end{aligned}$

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta-8$ and $\sec ^2\left(\tan ^{-1}…

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