Question : Simplify the given expression.
$\sqrt{\frac{1+\cos P}{1-\cos P}}$
Option 1: $\operatorname{cosec}P-\cot P$
Option 2: $\sec P-\tan P$
Option 3: $\sec P+\tan P$
Option 4: $\operatorname{cosec} P+\cot P$
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Correct Answer: $\operatorname{cosec} P+\cot P$
Solution :
Given: $\sqrt{\frac{1+\cos P}{1-\cos P}}$
Multiply the numerator and denominator by $1+\cos P$
$=\sqrt{\frac{(1+\cos P)^2}{(1-\cos P)(1+\cos P)}}$
$=\sqrt{\frac{(1+\cos P)^2}{(1-\cos^2 P)}}$
$=\sqrt{\frac{(1+\cos P)^2}{(\sin^2 P)}}$
$=\frac{(1+\cos P)}{(\sin P)}$
$=\operatorname{cosec}P + \cot P$
Hence, the correct answer is $\operatorname{cosec}P + \cot P$.
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