Question

Prove that: $2\cos \frac{A}{2}=\pm \sqrt{1-\sin A}\pm \sqrt{1+\sin A},$

Answer 1

$\frac{A}{2}={278}^{\circ }$, which is in fourth quadrant.

$\therefore \cos \frac{A}{2}$ is positive, $\sin \frac{A}{2}$ is negative.

$\mid \sin \frac{A}{2}\mid >\mid \cos \frac{A}{2}\mid$

$2\cos \frac{A}{2}=(\cos \frac{A}{2}+\sin \frac{A}{2})+(\cos \frac{A}{2}-\sin \frac{A}{2})$

$=\pm \sqrt{(\cos \frac{A}{2}+\sin \frac{A}{2}{)}^2}\pm \sqrt{(\cos \frac{A}{2}-\sin \frac{A}{2}{)}^2}$

$2\cos \frac{A}{2}=\pm \sqrt{1+\sin A}\pm \sqrt{1-\sin A}$