Question : If $\cos 5 \alpha=\sin \alpha$ and $5 \alpha<90^{\circ}$, then the value of $\tan 2 \alpha$ is:
Option 1: $\sqrt{2}$
Option 2: $\frac{1}{\sqrt{2}}$
Option 3: $\frac{1}{\sqrt{3}}$
Option 4: $\sqrt{3}$
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Correct Answer: $\frac{1}{\sqrt{3}}$
Solution :
Given, $\cos 5 \alpha=\sin \alpha$
We know, $\cos(90°-\theta)=\sin\theta$
⇒ $\cos5\alpha=\cos(90°-\alpha)$
⇒ $5\alpha=90°-\alpha$
⇒ $6\alpha=90°$
⇒ $\alpha=15°$
Now, $\tan 2 \alpha=\tan(2\times 15°)=\tan30°=\frac{1}{\sqrt3}$
Hence, the correct answer is $\frac{1}{\sqrt3}$.
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