MCQ
$\sqrt{\frac{1+\cos A}{1-\cos A}}= ?$
- A$\operatorname{cosec} A -\cot A$
- ✓$\operatorname{cosec} A+\cot A$
- C$\operatorname{cosec} A \cot A$
- Dnone of these
✓
Answer
Correct option: B.
$\operatorname{cosec} A+\cot A$
$\sqrt{\frac{1+\cos A }{1-\cos A }}$
$=\sqrt{\frac{(1+\cos A )}{(1-\cos A )} \times \frac{(1+\cos A )}{(1+\cos A )}}$
$=\frac{(1+\cos A )}{\sqrt{1-\cos ^2 A}}=\frac{1+\cos A }{\sqrt{\sin ^2 A}}$
$=\frac{1+\cos A }{\sin A }=\frac{1}{\sin A}+\frac{\cos A }{\sin A }$
$=\operatorname{cosec} A +\cot A $
$=\sqrt{\frac{(1+\cos A )}{(1-\cos A )} \times \frac{(1+\cos A )}{(1+\cos A )}}$
$=\frac{(1+\cos A )}{\sqrt{1-\cos ^2 A}}=\frac{1+\cos A }{\sqrt{\sin ^2 A}}$
$=\frac{1+\cos A }{\sin A }=\frac{1}{\sin A}+\frac{\cos A }{\sin A }$
$=\operatorname{cosec} A +\cot A $
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