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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