# If $\pi < \theta < \large\frac{3\pi}{2}$ the expression $\sqrt{4\sin^4\theta+\sin^22\theta}+4\cos^2\big(\large\frac{\pi}{4}-\frac{\theta}{2}\big)$ is equal to

$\begin{array}{1 1}(a)\;2&(b)\;2+4\sin\theta\\(c)\;2-4\sin\theta&(d)\;None\;of\;these\end{array}$

Given expression :
$\sqrt{4\sin^4\theta+\sin^2\theta}+4\cos^2\big(\large\frac{\pi}{2}-\frac{\theta}{2}\big)$
$\Rightarrow \sqrt{4\sin^4\theta+4\sin^2\theta\cos^2\theta}+2.2\cos^2\big(\large\frac{\pi}{2}-\frac{\theta}{2}\big)$
$\Rightarrow \sqrt{4\sin^2\theta(\sin^2\theta+\cos^2\theta)}+2\big[1+\cos 2\big(\large\frac{\pi}{2}-\frac{\theta}{2}\big)\big]$