Question : If $\cos \theta=\frac{\sqrt{3}}{2}$, then $\tan ^2 \theta \cos ^2 \theta=?$
Option 1: $\frac{1}{\sqrt{3}}$
Option 2: $\frac{1}{4}$
Option 3: $\frac{1}{2}$
Option 4: $\sqrt{3}$
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Correct Answer: $\frac{1}{4}$
Solution :
We have,
$\cos \theta = \frac{\sqrt{3}}{2}$
⇒ $\cos \theta = \cos 30^\circ$
⇒ $\theta = 30^\circ$
Now,
$\tan^2 \theta \cos^2 \theta=\tan^2 30^\circ \cos^2 30^\circ=
(\frac{1}{\sqrt3})^2 ×(\frac{\sqrt{3}}{2})^2$
$= \frac{1}{4}$
Hence, the correct answer is $\frac{1}{4}$.
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