Question : If $\sin \theta+\cos \theta = p$ and $\sec \theta + \operatorname{cosec} \theta = q$, then the value of $q \times (p^2-1)$ is:
Option 1: $1$
Option 2: $p$
Option 3: $2p$
Option 4: $2$
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Correct Answer: $2p$
Solution :
Given: $\sin \theta+\cos \theta = p$ and $\sec \theta + \operatorname{cosec} \theta = q$
Now, $q \times (p^2-1)$
Putting the values, we get:
= $(\sec \theta + \operatorname{cosec} \theta) [(\sin \theta+\cos \theta)^2 - 1]$
= $(\frac{1}{\cos \theta} + \frac{1}{\sin \theta})(2 \sin \theta \cos\theta)$
= $(\frac{\sin\theta + \cos\theta}{\sin \theta \cos\theta})(2 \sin \theta \cos\theta)$
= $2(\sin \theta +\cos\theta)$
= $2p$
Hence, the correct answer is $2p$.
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