Question : What is the value of the following in terms of trigonometric ratios?
$\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}$
Option 1: $2\operatorname{cosec A}$
Option 2: $2\operatorname{cos A}$
Option 3: $2\operatorname{sec A}$
Option 4: $2\operatorname{sin A}$
Correct Answer: $2\operatorname{cosec A}$
Solution :
$\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}$
$=\frac{\sin^2 A + (1 + \cos A)^2}{\sin A (1 + \cos A)}$
$=\frac{\sin^2 A +1 + 2\cos A + \cos^2 A}{\sin A (1 + \cos A)}$
$=\frac{2 + 2\cos A}{\sin A (1 + \cos A)}$ [$\because \sin^2 A + \cos^2 A = 1$]
$=\frac{2(1 + \cos A)}{\sin A (1 + \cos A)}$
$=\frac{2}{\sin A}$
$=2\operatorname{cosec A}$
Hence, the correct answer is $2\operatorname{cosec A}$.
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