First slide

Multiple and sub- multiple Angles

Question

If $2 \cos \frac{A}{2} = \sqrt{1 + \sin A} + \sqrt{1 - \sin A},$ then $\frac{A}{2}$

Moderate

A

$2 n π + \frac{π}{4}$ and $2 n π + \frac{3 π}{4}$
B

$2 n π - \frac{π}{4}$ and $2 n π + \frac{π}{4}$
C

$2 n π - \frac{3 π}{4}$ and $2 n π - \frac{π}{4}$
D

$- \infty and + \infty$

Solution

We have,
$\begin{array}{l} 2 \cos \frac{A}{2} = \sqrt{1 + \sin A} + \sqrt{1 - \sin A} \\ \Rightarrow 2 \cos \frac{A}{2} = \sqrt{{(\cos \frac{A}{2} + \sin \frac{A}{2})}^{2}} + \sqrt{{(\cos \frac{A}{2} - \sin \frac{A}{2})}^{2}} \\ \Rightarrow 2 \cos \frac{A}{2} = |\cos \frac{A}{2} + \sin \frac{A}{2}| + |\cos \frac{A}{2} - \sin \frac{A}{2}| \\ \Rightarrow \cos \frac{A}{2} + \sin \frac{A}{2} \geq 0 and \cos \frac{A}{2} - \sin \frac{A}{2} \geq 0 \\ \Rightarrow - \frac{3 π}{4} \leq \frac{A}{2} \leq \frac{3 π}{4} and - \frac{π}{4} \leq \frac{A}{2} \leq \frac{π}{4} \\ \Rightarrow - \frac{π}{4} \leq \frac{A}{2} \leq \frac{π}{4} \end{array}$
$\Rightarrow 2 n π - \frac{π}{4} \leq \frac{A}{2} \leq 2 n π + \frac{π}{4}, n \in Z$

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