Question

Prove that $-2\sin \frac{A}{2}=-\sqrt{1+\sin A}-\sqrt{1-\sin A}$ ?

Answer 1

$RHS=-\sqrt{1+\sin A}-\sqrt{1-\sin A}=-\sqrt{1+\sin 2.\frac{A}{2}}-\sqrt{1-\sin 2.\frac{A}{2}}=-\sqrt{{\sin }^2\frac{A}{2}+{\cos }^2\frac{A}{2}+2\sin \frac{A}{2}\cos \frac{A}{2}}-\sqrt{{\sin }^2\frac{A}{2}+{\cos }^2\frac{A}{2}-2\sin \frac{A}{2}\cos \frac{A}{2}}(\because \sin 2\theta =2\sin \theta \cos \theta {\sin }^2\theta +{\cos }^2\theta =1)=-\sqrt{{(\sin \frac{A}{2}+\cos \frac{A}{2})}^2}-\sqrt{{(\sin \frac{A}{2}-\cos \frac{A}{2})}^2}=-(\sin \frac{A}{2}+\cos \frac{A}{2})-(\sin \frac{A}{2}-\cos \frac{A}{2})=-2\sin \frac{A}{2}=LHS$