If 180^{circ}

Question

(A) Given expression: $E = \sqrt{4 \sin^{4} 	heta + \sin^{2} 2	heta} + 4 \cos^{2} \left(\frac{\pi}{4} - \frac{	heta}{2}ight)$.Using $\sin 2	heta

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$2$

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(A) Given expression: $E = \sqrt{4 \sin^{4} 	heta + \sin^{2} 2	heta} + 4 \cos^{2} \left(\frac{\pi}{4} - \frac{	heta}{2}ight)$.Using $\sin 2	heta = 2 \sin 	heta \cos 	heta,$ we have $\sin^{2} 2	heta = 4 \sin^{2} 	heta \cos^{2} 	heta$.So,$\sqrt{4 \sin^{4} 	heta + 4 \sin^{2} 	heta \cos^{2} 	heta} = \sqrt{4 \sin^{2} 	heta (\sin^{2} 	heta + \cos^{2} 	heta)} = \sqrt{4 \sin^{2} 	heta} = 2 |\sin 	heta|$.Using $2 \cos^{2} A = 1 + \cos 2A,$ the second term is $2 [1 + \cos(\frac{\pi}{2} - 	heta)] = 2(1 + \sin 	heta)$.Thus,$E = 2 |\sin 	heta| + 2 + 2 \sin 	heta$.Since $180^{\circ} Therefore,$E = 2(-\sin 	heta) + 2 + 2 \sin 	heta = 2$.

If $180^{\circ} < \theta < 270^{\circ},$ then the value of $\sqrt{4 \sin^{4} \theta + \sin^{2} 2\theta} + 4 \cos^{2} \left(\frac{\pi}{4} - \frac{\theta}{2}\right)$ is

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