If <x<2 , then 1+ 1- is equal to: - Vidyadip

MCQ

If $\pi<\text{x}<2\pi,$ then $\sqrt{\frac{1+\cos\text{x}}{1-\cos\text{x}}}$ is equal to:

A
$\text{cosec x}+\cot \text{x}$
B
$\text{cosec x}-\cot \text{x}$
C
$-\text{cosec x}+\cot \text{x}$
✓
$-\text{cosec x}-\cot \text{x}$

✓

Answer

Correct option: D.

$-\text{cosec x}-\cot \text{x}$

$-\text{cosec}\text{x} -\cot\text{x}$
$=\sqrt{\frac{1+\cos\text{x}}{1-\cos\text{x}}}$
$=\sqrt{\frac{(1+\cos\text{x})(1+\cos\text{x})}{(1+\cos\text{x})(1+\cos\text{x})}}$
$=\sqrt{\frac{(1+\cos\text{x})^2}{1-\cos^2\text{x}}}$
$=\sqrt{\frac{(1+\cos\text{x})^2}{\sin^2\text{x}}}$
$=\frac{(1+\cos\text{x})}{-\sin\text{x}}$ $ [\text{as},\pi<\text{x}<2\pi,\text{ so }\sin\text{x} \text{ will}\text{ be}\text{ negative}]$
$=-(\text{cosec}\text{ x}+\cot\text{x})$
$= -\text{cosec}\text{ x } -\cot\text{x}$

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