Question : If $\theta$ is a positive acute angle and $4\sin^{2}\theta =3$, then the value of $\left (\tan\theta-\cot\frac{\theta}{2}\right)$ is:
Option 1: $1$
Option 2: $0$
Option 3: $\sqrt{3}$
Option 4: $\frac{1}{\sqrt{3}}$
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Correct Answer: $0$
Solution :
Given: $4\sin^{2}\theta =3$
$⇒\sin\theta = \pm\frac{\sqrt{3}}{2}=\sin 60°$
Since $\theta$ is acute, so, $\theta=60°$
So, $\tan\theta-\cot\frac{\theta}{2}=\tan60°-\cot30°=\sqrt3-\sqrt3=0$
Hence, the correct answer is 0.
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