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