Question

If $y = \sqrt{\left(\frac{1+\cos 2\theta}{1 -\cos 2\theta}\right)} $ , then $\frac{dy}{d\theta} $ at $\theta =\frac{3\pi}{4}$ is:

• A. -2
• B. 2
• C. $\pm$
• D. none of these

Answer 1

Answer: (A)

$y = \sqrt{\left(\frac{1+\cos 2\theta}{1 -\cos 2\theta}\right)} $ ,
$\Rightarrow \:\: y = \sqrt{ \frac{2 \:\cos^2 \theta}{2 \: \sin^2\theta}} = \sqrt{\cot^2 \: \theta}$
$\Rightarrow \:\: y = \cot \: \theta$
Differentiate w.r.t. $' \theta '$, we get
$\frac{dy }{d \theta} = - cosec^2 \theta$
Now , $ \left( \frac{dy }{d \theta}\right)_{\theta = \frac{3 \pi}{4}}$
$ = -cosec^2 \left( \frac{3 \pi}{4} \right)$
$ = -cosec^2 \left(\pi - \frac{ \pi}{4} \right)$
$ = -cosec^2 \frac{ \pi}{4}$
$ = - 2 \left( \because \sin \frac{ \pi}{4} = \frac{ 1}{\sqrt{2}} \right) $