Question : If $\cos \theta+\cos ^2 \theta=1$, find the value of $\sqrt{\sin ^4 \theta+\cos ^2 \theta}$.
Option 1: $\sqrt{2} \cos \theta$
Option 2: $2 \operatorname{cos} \theta$
Option 3: $\sqrt{2} \operatorname{sin} \theta$
Option 4: $2 \operatorname{sin} \theta$
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Correct Answer: $\sqrt{2} \cos \theta$
Solution :
Given: $\cos \theta+\cos ^2 \theta=1$
⇒ $\cos\theta + \cos^2\theta = \sin^2\theta + \cos^2\theta$
⇒ $\cos\theta = \sin^2\theta$
Putting the value in the given equation,
$\sqrt{\sin ^4 \theta+\cos ^2 \theta}$
$=\sqrt{\cos^2\theta + \cos^2\theta}$
$=\sqrt{2} \cos\theta$
Hence, the correct answer is $\sqrt{2} \cos\theta.$
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