The minimum value of left(1+frac{1}{sin ^n al | vedclass

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(D) Let $f(\alpha) = \left(1 + \frac{1}{\sin^n \alpha}\right) \left(1 + \frac{1}{\cos^n \alpha}\right)$.By the $AM$-$GM$ inequality,for positive terms

Vedclass · Accepted Answer

$\left(1+2^{n / 2}\right)^2$

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(D) Let $f(\alpha) = \left(1 + \frac{1}{\sin^n \alpha}\right) \left(1 + \frac{1}{\cos^n \alpha}\right)$.By the $AM$-$GM$ inequality,for positive terms,the minimum value occurs when $\sin^n \alpha = \cos^n \alpha$,which implies $\sin \alpha = \cos \alpha$,so $\alpha = \frac{\pi}{4}$.At $\alpha = \frac{\pi}{4}$,we have $\sin \alpha = \cos \alpha = \frac{1}{\sqrt{2}} = 2^{-1/2}$.Then $\sin^n \alpha = \cos^n \alpha = (2^{-1/2})^n = 2^{-n/2}$.Substituting this into the expression:$f\left(\frac{\pi}{4}\right) = \left(1 + \frac{1}{2^{-n/2}}\right) \left(1 + \frac{1}{2^{-n/2}}\right) = (1 + 2^{n/2})(1 + 2^{n/2}) = (1 + 2^{n/2})^2$.Thus,the minimum value is $(1 + 2^{n/2})^2$.

The minimum value of $\left(1+\frac{1}{\sin ^n \alpha}\right)\left(1+\frac{1}{\cos ^n \alpha}\right)$ is

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