Question

If $y=1-\cos \theta ,x=1-\sin \theta$, then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$ is

• A. $-1$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A $-1$

Given, $y=1-\cos \theta$

$\Rightarrow \frac{dy}{d\theta }=\sin \theta$
Also given, $x=1-\sin \theta$

$\Rightarrow \frac{dx}{d\theta }=-\cos \theta$
Therefore, $\frac{dy}{dx}=\frac{\sin \theta }{-\cos \theta }=-\tan \theta$ at $\theta =\frac{\pi }{4}$ will be
$\therefore {\left[\frac{dy}{dx}\right]}_{\theta =\frac{\pi }{4}}=-\tan \frac{\pi }{4}=-1$