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  4. If x ∈(π, (3 π/2)), then 4 cos 2((π/4)-(x/2))+√4 sin 4 x+ sin 2 2 x is always equal to

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Q. If $x \in\left(\pi, \frac{3 \pi}{2}\right)$, then $4 \cos ^{2}\left(\frac{\pi}{4}-\frac{x}{2}\right)+\sqrt{4 \sin ^{4} x+\sin ^{2} 2 x}$ is always equal to

Trigonometric Functions

Solution:

$4 \cos ^{2}\left(\frac{\pi}{4}-\frac{x}{2}\right)+\sqrt{4 \sin ^{4} x+\sin ^{2} 2 x}$
$=4 \cos ^{2}\left(\frac{\pi}{4}-\frac{x}{2}\right)+\sqrt{4 \sin ^{2} x\left(\cos ^{2} x+\sin ^{2} x\right)}$
$=2\left(1+\cos \left(\frac{\pi}{2}-x\right)\right)+2|\sin x|$
$=2+2 \sin x-2 \sin x$ as $x \in\left(\pi, \frac{3 \pi}{2}\right)=2$