If π < x <2π,
Question:

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

(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

$\sqrt{\frac{1+\cos x}{1-\cos x}}$

$=\sqrt{\frac{(1+\cos x)(1+\cos x)}{(1-\cos x)(1+\cos x)}}$

$=\sqrt{\frac{(1+\cos x)^{2}}{1-\cos ^{2} x}}$

 

$=\sqrt{\frac{(1+\cos x)^{2}}{\sin ^{2} x}}$

$=\frac{(1+\cos x)}{-\sin x} \quad[\operatorname{as}, \pi<x<2 \pi$, so $\sin x$ will be negative $]$

$=-(\operatorname{cosec} x+\cot x)$

 

$=-\operatorname{cosec} x-\cot x$

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