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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