Question

If $\sqrt{\frac{1-\sin A}{1+\sin A}}+\frac{\sin A}{\cos A}=\frac{1}{\cos A}$, for all permissible values of $A$, then $A$ belongs to

• A. First Quadrant
• B. Second Quadrant
• C. Third Quadrant
• D. Fourth Quadrant

Answer 1

Answer: (A), (D)

First Quadrant
Fourth Quadrant

$\sqrt{\frac{1-\sin A}{1+\sin A}}+\frac{\sin A}{\cos A}=\frac{1}{\cos A}$
Hence
$|\frac{1-\sin A}{\cos A}|=\frac{1-\sin A}{\cos A}$.
Now
$-1\le \sin A\le 1$
Or
$0\le 1-\sin A\le 2$.
Hence
$|1-\sin A|=1-\sin A.$
Hence we get
$(1-\sin A)(\frac{1}{|\cos A|}-\frac{1}{\cos A})=0$
Or
$\sin A=1$ and $A=\frac{\pi }{2}$ or
$|\cos A|=\cos A$. This implies $Aϵ[-\frac{\pi }{2},\frac{\pi }{2}]$.
Hence Ist and IVth quadrant.