Question

If $x=a\left[\cos t+\frac{1}{2}\log \left({\tan }^2\frac{t}{2}\right)\right]$ and $y=a\sin t$, then find $\frac{dy}{dx}$ at $t=\frac{\pi }{4}$.

Answer 1

Answer: 1

$x=a\left[\cos t+\frac{1}{2}\log \left({\tan }^2\frac{t}{2}\right)\right]$
$\Rightarrow x=a\left[\cos t+\log \left(\tan \frac{t}{2}\right)\right]$
$\Rightarrow \frac{dx}{dt}=a\left(-\sin t+\frac{1}{\tan \frac{t}{2}}\cdot {\sec }^2\frac{t}{2}\cdot \frac{1}{2}\right)$
$=a\left(-\sin t+\frac{1}{2\sin \frac{t}{2}\cos \frac{t}{2}}\right)$
$=a\left(-\sin t+\frac{1}{\sin t}\right)$
$=a\left(\frac{1-{\sin }^{2}t}{\sin t}\right)$
$=a\left(\frac{{\cos }^{2}t}{\sin t}\right)$
$y=a\sin t$
$\Rightarrow \frac{dy}{dt}=a\cos t$
$\Rightarrow \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{a\cos t}{\frac{a{\cos }^{2}t}{\sin t}}=\tan t$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{t=\frac{\pi }{4}}=\tan \frac{\pi }{4}=1$