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

The Correct Option is 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$