Question

Demonstrati ca:
$\sqrt{1+sinx} -\sqrt{1-sinx} =2sin\frac{x}{2} ; x apartine (0,\frac{\pi }{2} )$[/tex]

Answer 1

$E = \sqrt{1+\sin x} - \sqrt{1-\sin x} \\ E^2 = 1+\sin x +1-\sin x -2\sqrt{1-\sin^2 x} \\ E^2 = 2-2\sqrt{\cos^2 x} \\ E^2 = 2-2\cos x  \\ E^2 = 2(1-\cos x) |:4 \\ \dfrac{E^2}{4} = \dfrac{1-\cos x}{2} \\ \\ E^2 = 4\cdot \sqrt{\dfrac{1- \cos x}{2}} ~~\Big| \sqrt{} \\ \\ E = 2\sin \frac{x}{2},\quad x \in\Big(0,\dfrac{\pi}{2}\Big)$