First slide

Multiple and sub- multiple Angles

Question

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

Moderate

A

1
B

2
C

-2
D

none of these

Solution

$\begin{matrix} 4 \cos^{2} (\frac{π}{4} - \frac{x}{2}) + \sqrt{4 \sin^{4} x + \sin^{2} 2 x} \\ = 4 \cos^{2} (\frac{π}{4} - \frac{x}{2}) + \sqrt{4 \sin^{2} x (\cos^{2} x + \sin^{2} x)} \\ = 2 (1 + \cos (\frac{π}{2} - x)) + 2 | \sin x | \\ = 2 + 2 \sin x - 2 \sin x \\ = 2 \end{matrix}$

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