Question

$x=\sin \theta \sqrt{\cos 2\theta },y=\cos \theta \sqrt{\sin 2\theta },$ find $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}.$Insert 0 if the value is not defined and 1 otherwise.

Answer 1

$x=\sin \theta \sqrt{\cos 2\theta },y=\cos \theta \sqrt{\sin 2\theta },$

$\Rightarrow \log x=\log \sin \theta +\frac{1}{2}\log \cos 2\theta ,\log y=\log \cos \theta +\frac{1}{2}\log \sin 2\theta$

Differentiating w.r.t $\theta$

$\Rightarrow \frac{1}{x}\frac{dx}{d\theta }=\cot \theta -\tan 2\theta ,\frac{1}{y}\frac{dy}{d\theta }=-\tan \theta +\cot 2\theta$

$\therefore \frac{dy}{dx}=\frac{y(-\tan \theta +\cot 2\theta )}{x(\cot \theta -\tan 2\theta )}$

Clearly at $\theta =\frac{\pi }{4}$, $dy/dx$ is not defined