Question

If $x={\cos }^2\theta$ and $y=\cot \theta$, then find $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$

Answer 1

Given

$x={\cos }^2\theta ,y=\cot \theta$

$dx=-2\cos \theta \sin \theta d\theta$

$dy=-{\text{cosec}}^2\theta d\theta$

$\frac{dy}{dx}=\frac{-{\text{cosec}}^2\theta d\theta }{-2\cos \theta \sin \theta d\theta }$

$=\frac{1}{2{\sin }^2\theta \cos \theta \sin \theta }$

$\therefore {\left(\frac{dy}{dx}\right)}_{\theta =\frac{\pi }{4}}=\frac{1}{\frac{1}{2\sqrt{2}}\times \frac{1}{\sqrt{2}}\times 2}=2$