If theta in left(frac{pi}{2}, frac{3 pi}{2}ri | vedclass

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(B) Given expression: $\sqrt{4 \cos^{4} 	heta + \sin^{2} 2 	heta} + 4 \cot 	heta \cos^{2} \left(\frac{\pi}{4} - \frac{	heta}{2}ight)$$= \sqrt{4

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$2 \cot 	heta$

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(B) Given expression: $\sqrt{4 \cos^{4} 	heta + \sin^{2} 2 	heta} + 4 \cot 	heta \cos^{2} \left(\frac{\pi}{4} - \frac{	heta}{2}ight)$$= \sqrt{4 \cos^{4} 	heta + (2 \sin 	heta \cos 	heta)^{2}} + 2 \cot 	heta \left[2 \cos^{2} \left(\frac{\pi}{4} - \frac{	heta}{2}ight)ight]$$= \sqrt{4 \cos^{4} 	heta + 4 \sin^{2} 	heta \cos^{2} 	heta} + 2 \cot 	heta \left[1 + \cos \left(\frac{\pi}{2} - 	hetaight)ight]$$= \sqrt{4 \cos^{2} 	heta (\cos^{2} 	heta + \sin^{2} 	heta)} + 2 \cot 	heta (1 + \sin 	heta)$$= |2 \cos 	heta| + 2 \cot 	heta + 2 \cot 	heta \sin 	heta$$= |2 \cos 	heta| + 2 \cot 	heta + 2 \cos 	heta$Since $	heta \in \left(\frac{\pi}{2}, \frac{3 \pi}{2}ight)$,$\cos 	heta $= -2 \cos 	heta + 2 \cot 	heta + 2 \cos 	heta$$= 2 \cot 	heta$

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

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