If `x in (pi,(3pi)/2),`then `4cos^2(pi/4-x/2)+sqrt(4sin^4x+sin^2 2x)` is always equal to (a) 1 (b) 2 (c) `-2` (d) none of these

Explanation

To solve the expression \( 4\cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right) + \sqrt{4\sin^4x + \sin^2 2x} \) for \( x \) in the interval \( \left(\pi, \frac{3\pi}{2}\right) \), we will break it down step by step.

### Step 1: Rewrite the Expression
We start with the expression:
\[ 4\cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right) + \sqrt{4\sin^4x + \sin^2 2x} \]

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