Question

If $x$ and $y$ are connected parametrically by the given equation, then without eliminating the parameter, find $\frac{dy}{dx}$.
$x=a(\cos t+\log \tan \frac{t}{2}),y=a\sin t$

Answer 1

The given equations are $x=a(\cos t+\log \tan \frac{t}{2}),y=a\sin t$
Then, $\frac{dx}{dt}=a\left[\frac{d}{dt}\right(\cos t)+\frac{d}{dt}\left(\log \tan \frac{t}{2}\right)]$
$=a[-\sin t+\frac{1}{\tan \frac{t}{2}}.\frac{d}{dt}\left(\tan \frac{t}{2}\right)]$
$=a[-\sin t+\cot \frac{t}{2}.{\sec }^2\frac{t}{2}.\frac{d}{dt}\left(\frac{t}{2}\right)]$
$=a[-\sin t+\frac{\cos \frac{t}{2}}{\sin \frac{t}{2}}\times \frac{1}{{\cos }^2\frac{t}{2}}\times \frac{1}{2}]$
$=a[-\sin t+\frac{1}{2\sin \frac{t}{2}\cos \frac{t}{2}}]$
$=a\left(\frac{-{\sin }^{2}t+1}{\sin t}\right)=a\frac{{\cos }^{2}t}{\sin t}$
& $\frac{dy}{dt}=a\frac{d}{dt}\left(\sin t\right)=a\cos t$
$\therefore \frac{dy}{dx}=\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}=\frac{a\cos t}{\left(a\frac{{\cos }^{2}t}{\sin t}\right)}=\frac{\sin t}{\cos t}=\tan t$