Question

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

• A. $cosec\theta +cot\theta$
• B. $cosec\theta -cot\theta$
• C. $-cosec\theta +cot\theta$
• D. $-cosec\theta -cot\theta$

Answer 1

The correct option is D

$-cosec\theta -cot\theta$

$-cosec\theta -cot\theta$

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

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

$=\sqrt{\frac{(1+cos\theta {)}^2}{1-co{s}^2\theta }}$

$=\sqrt{\frac{(1+cos\theta {)}^2}{si{n}^2\theta }}$

$=\frac{(1+cos\theta )}{-sin\theta }$

[as, $\pi <2\pi$, so $sin\theta$ will be negative]

$=-(cosec\theta +cot\theta )$

$=-cosec\theta -cot\theta$