Question : If $\tan \theta \cdot \tan 2 \theta=1$, then the value of $\cot 5 \theta$ is:
Option 1: $-1$
Option 2: $1$
Option 3: $-\sqrt{3}$
Option 4: $\sqrt{3}$
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Correct Answer: $-\sqrt{3}$
Solution :
Given: $\tan \theta \cdot \tan 2 \theta=1$
$⇒\tan2\theta=\cot\theta$
$⇒\tan2\theta=\tan(90°- \theta)$
$⇒2\theta=90°- \theta$
$⇒3\theta=90°$
$\therefore \theta=30°$
$\cot 5\theta=\cot(5×30°)=\cot 150°=\cot(90°+60°)=-\tan60°=-\sqrt3$
Hence, the correct answer is $-\sqrt3$.
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