Question : Find the value of $\tan 3 \theta$, if $\sec 3 \theta=\operatorname{cosec}\left (4 \theta-15°\right)$.
Option 1: $\frac{1}{\sqrt{3}}$
Option 2: $\sqrt3$
Option 3: $–1$
Option 4: $1$
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Correct Answer: $1$
Solution :
Given, $\sec 3 \theta=\operatorname{cosec}\left (4 \theta-15°\right)$
We know $ \operatorname{cosec}\theta = \sec(90°-\theta)$
⇒ $\sec3\theta = \sec (90°-(4\theta - 15°))$
⇒ $3\theta = 90°-4\theta + 15°$
⇒ $7\theta = 105°$
⇒ $\theta = 15°$
$\therefore$ $\tan 3 \theta=\tan (3\times15)=\tan 45° = 1$
Hence, the correct answer is $1$.
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