First slide

Multiple and sub- multiple Angles

Question

If $π < α < \frac{3 π}{2},$ the the expression $\sqrt{4 \sin^{4} α + \sin^{2} 2 α} + 4 \cos^{2} (\frac{π}{4} - \frac{α}{2})$ is equal to

Moderate

A

$2 + 4 \sin α$
B

$2 - 4 \sin α$
C

2
D

none of these

Solution

We have,
$\begin{array}{l} \sqrt{4 \sin^{4} α + \sin^{2} 2 α + 4 \cos^{2} (\frac{π}{4} - \frac{α}{2})} \\ = \sqrt{4 \sin^{2} α (\sin^{2} α + \cos^{2} α) + 4} \{\frac{1 + \cos (\frac{π}{2} - α)}{2}\} \\ = 2 | \sin α | + 2 (1 + \sin α) \\ = - 2 \sin α + 2 (1 + \sin α) = 2 [\begin{array}{l} ∵ \sin α < 0] \\ for α \in 3 π / 2 \end{array}] \end{array}$

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