Question

If $x=a\left(\cos t+\log \tan \frac{t}{2}\right),y=a\sin t,$ then $\frac{dy}{dx}$

• A. $\tan t$
• B. $\cot t$
• C. $-\cot t$
• D. $-\tan t$

Answer 1

Answer: (A)

We have, $x=a\left(\cos t+\log \tan \frac{t}{2}\right)$
$\Rightarrow \frac{dy}{dx}=-a\sin t+a\frac{\frac{1}{2}{\sec }^2\left(t/2\right)}{\tan \left(t/2\right)}$
$y=a\sin t\Rightarrow \frac{dy}{dt}=a\cos t$
$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{a\cos t}{a\left(-\sin t+\frac{{\sec }^2\left(t/2\right)}{2\tan \left(t/2\right)}\right)}$
$=\frac{a\cos t}{a\left[-\sin t+\frac{1}{\sin t}\right]}$
$=\frac{\cos t\sin t}{-\sin t+1}=\frac{\cos t\sin t}{{\cos }^{2}t}=\tan t$