Question : If $\sin^2 \theta \cos^2 \theta=\frac{2}{9}$, then what will be the value of $\operatorname{cosec}^2 \theta+\sec ^2 \theta$?
Option 1: $\frac{7}{2}$
Option 2: $\frac{5}{2}$
Option 3: $\frac{9}{2}$
Option 4: $9 \sqrt{2}$
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Correct Answer: $\frac{9}{2}$
Solution :
Given, $\sin^2 \theta \cos^2 \theta=\frac{2}{9}$
Now, $\operatorname{cosec}^2 \theta+\sec ^2 \theta = \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta}$
$⇒\operatorname{cosec}^2 \theta+\sec ^2 \theta = \frac{\sin^2 \theta+\cos^2 \theta}{\sin^2 \theta \cos^2 \theta}$
We know that $\sin^2 \theta+\cos^2 \theta=1$
So, $\operatorname{cosec}^2 \theta+\sec ^2 \theta = \frac{1}{\frac{2}{9}}$
$\therefore\operatorname{cosec}^2 \theta+\sec ^2 \theta = \frac{9}{2}$
Hence, the correct answer is $\frac{9}{2}$.
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