Question : Find the numerical value of
$\frac{9}{\operatorname{cosec^{2}\theta}}+4\cos^{2}\theta+\frac{5}{1+\tan^{2}\theta}$
Option 1: $5$
Option 2: $7$
Option 3: $9$
Option 4: $4$
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Correct Answer: $9$
Solution :
Given: $\frac{9}{\operatorname{cosec^{2}\theta}}+4\cos^{2}\theta+\frac{5}{1+\tan^{2}\theta}$
$=9\sin^2\theta+4\cos^{2}\theta+\frac{5}{\operatorname{sec^{2}\theta}}$
$=9\sin^2\theta+4\cos^{2}\theta+5\cos^{2}\theta$
$=9\sin^2\theta+9\cos^{2}\theta$
$=9(\sin^2\theta+\cos^{2}\theta)$
$=9$ [$\because \sin^2\theta+\cos^{2}\theta=1$]
Hence, the correct answer is $9$.
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