Question

If $x=\left(\cos \theta +\log \tan \theta \right)$ and $y=\sin \theta$,Find the value of $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$

Answer 1

Given $y=\sin \theta$ and $x=\cos \theta +\tan \theta$

Since,

$\frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$...........(i)

Now,

$y=\sin \theta$

$\frac{dy}{d\theta }=\cos \theta$........(ii)

$x=\cos \theta +\log \tan \theta$

$\frac{dx}{d\theta }=-\sin \theta +\frac{{\sec }^2\theta }{\tan \theta }=$

Hence,

$\frac{dy}{dx}=\frac{\cos \theta }{-\sin \theta +\frac{{\sec }^2\theta }{\tan \theta }}$

$\frac{dy}{dx}{|}_{x=\frac{\pi }{4}}=\frac{1/\sqrt{2}}{-1/\sqrt{2}+\frac{2}{1}}=\frac{1}{2\sqrt{2}-1}$