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. ${}^{\prime }{\theta }^{\prime }$, we get
$\frac{dy}{d\theta }=-cose{c}^2\theta$
Now , ${\left(\frac{dy}{d\theta }\right)}_{\theta =\frac{3\pi }{4}}$
$=-cose{c}^2\left(\frac{3\pi }{4}\right)$
$=-cose{c}^2\left(\pi -\frac{\pi }{4}\right)$
$=-cose{c}^2\frac{\pi }{4}$
$=-2\left(\because \sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}\right)$