Question:
If $\pi (a) cosec x + cot x (b) cosec x − cot x (c) −cosec x + cot x (d) −cosec x − cot x
Solution:
(d) −cosec x − cot x
$\sqrt{\frac{1+\cos x}{1-\cos x}}$
$=\sqrt{\frac{(1+\cos x)(1+\cos x)}{(1-\cos x)(1+\cos x)}}$
$=\sqrt{\frac{(1+\cos x)^{2}}{1-\cos ^{2} x}}$
$=\sqrt{\frac{(1+\cos x)^{2}}{\sin ^{2} x}}$
$=\frac{(1+\cos x)}{-\sin x} \quad[\operatorname{as}, \pi
$=-(\operatorname{cosec} x+\cot x)$
$=-\operatorname{cosec} x-\cot x$
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