Question

If $\alpha \in [\frac{\pi }{2},\pi ]$ then the value of $\sqrt{1+\sin \alpha }-\sqrt{1-\sin \alpha }$ is equal to:

• A. $2\cos \frac{\alpha }{2}$
• B. $2\sin \frac{\alpha }{2}$
• C. $2$
• D. none of these

Answer 1

The correct option is A $2\cos \frac{\alpha }{2}$

we have, $\sqrt{1+sin\alpha }-\sqrt{1-sin\alpha }$

$={\left(\sqrt{sin\frac{\alpha }{2}+cos\frac{\alpha }{2}}\right)}^2-{\left(\sqrt{sin\frac{\alpha }{2}+cos\frac{\alpha }{2}}\right)}^2$

$=sin\frac{\alpha }{2}+cos\frac{\alpha }{2}-sin\frac{\alpha }{2}+cos\frac{\alpha }{2}$

$=2cos\frac{\alpha }{2}$