Question

If $\cos \left(A/2\right)=\sqrt{(1+\sin A)}-\sqrt{(1-\sin A)}$, then

• A. $n\pi +\left(\frac{\pi }{4}\right)<\frac{A}{2}<n\pi +\left(\frac{3\pi }{4}\right)$
• B. $n\pi -\left(\frac{\pi }{4}\right)<\frac{A}{2}<2n\pi +\left(\frac{3\pi }{4}\right)$
• C. $2n\pi -\left(\frac{3\pi }{4}\right)<\frac{A}{2}<2n\pi +\left(\frac{5\pi }{4}\right)$
• D. $2n\pi +\left(\frac{\pi }{4}\right)<\frac{A}{2}<2n\pi +\left(\frac{3\pi }{4}\right)$

Answer 1

The correct option is D $2n\pi +\left(\frac{\pi }{4}\right)<\frac{A}{2}<2n\pi +\left(\frac{3\pi }{4}\right)$

We know that
$\sin \left(\frac{A}{2}\right)+\cos \left(\frac{A}{2}\right)=\pm \sqrt{(1+\sin A)}$
and $\sin \left(\frac{A}{2}\right)-\cos \left(\frac{A}{2}\right)=\pm \sqrt{(1-\sin A)}$
But we are given,
$2\cos \left(\frac{A}{2}\right)=\sqrt{(1+\sin A)}-\sqrt{(1-\sin A)}$
Now $3$ will be obtained if we take positive sing both in $\left(1\right)$ and $\left(2\right)$. But we know that
$\sin \left(\frac{A}{2}\right)+\cos \left(\frac{A}{2}\right)$ and $\sin \left(\frac{A}{2}\right)-\cos \left(\frac{A}{2}\right)$ are both positive when $\frac{\pi }{4}<\frac{A}{2}<\frac{3\pi }{4}$ or in general $2n\pi +\frac{\pi }{4}<\frac{A}{2}<2n\pi +\frac{3\pi }{4}$