If, for some , with and +=8, ^2(^-1)+cosec^2(^-1)=36, then ^2+ is ____.

Question

If, for some $\alpha,\beta$ with $\alpha\le\beta$ and $\alpha+\beta=8$, $\sec^2(	an^{-1}\alpha)+	ext{cosec}^2(\cot^{-1}\beta)=36$, then $\alpha^2+\beta$ is \_\_\_\_.

Answer: 14

Accepted Answer

$\sec^2(\tan^{-1}\alpha)=1+\alpha^2$ and $\text{cosec}^2(\cot^{-1}\beta)=1+\beta^2$, so

$$2+\alpha^2+\beta^2=36\ \Rightarrow\ \alpha^2+\beta^2=34.$$

Since $\alpha+\beta=8$: $(\alpha+\beta)^2=64=\alpha^2+\beta^2+2\alpha\beta=34+2\alpha\beta\Rightarrow\alpha\beta=15$.

So $\alpha,\beta$ are roots of $t^2-8t+15=0\Rightarrow t=3,5$. Since $\alpha\le\beta$: $\alpha=3,\ \beta=5$.

$$\alpha^2+\beta=9+5=14.$$

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