Question

If $\theta \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$, then the value of $\sqrt{4 \cos ^{4} \theta+\sin ^{2} 2 \theta}+4 \cot \theta \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)$

• A. $-2 \cot \theta$
• B. $2 \cot \theta$
• C. $2 \cos \theta$
• D. $2 \sin \theta$

Answer 1

The Correct Option is B

$\sqrt{4 \cos ^{4} \theta+\sin ^{2} 2 \theta}+4 \cot \theta \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)$
$=\sqrt{4 \cos ^{4} \theta+(2 \sin \theta \cos \theta)^{2}}$
$+2 \cot \theta\left[2 \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)\right]$
$=\sqrt{4 \cos ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta}$
$+2 \cot \theta\left[1+\cos \left(\frac{\pi}{2}-\theta\right)\right]$
$\left[\because 2 \cos ^{2} \theta=1+\cos 2 \theta\right]$
$=\sqrt{4 \cos ^{2} \theta\left(\cos ^{2} \theta+\sin ^{2} \theta\right)}+2 \cot \theta(1+\sin \theta)$
$=|2 \cos \theta|+2 \cot \theta+2 \cos \theta$
$=-2 \cos \theta+2 \cot \theta+2 \cos \theta$
$\left[\text{for} \theta \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right), \cos \theta \mid=-\cos \theta\right]$
$=2 \cot \theta$