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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