Question

If the parametric equation of a curve is given by $x=\cos \theta +\log \tan \frac{\theta }{2}$ and $y=\sin \theta$, then the points for which $\frac{dy}{dx}=0$ are given by

• A. $\theta =\frac{n\pi }{2},n\in z$
• B. $\theta =(2n+1)\frac{\pi }{2},n\in z$
• C. $\theta =(2n+1)\pi ,n\in z$
• D. $\theta =n\pi ,n\in z$

Answer 1

Answer: (D)

$\frac{dx}{d0}=-\sin \theta +\frac{1}{\tan \left(\frac{\theta }{2}\right)}\cdot {\sec }^2\left(\frac{\theta }{2}\right)\frac{1}{2}$
$=-\sin \theta +\frac{1}{2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)}$
$=-\sin \theta +\frac{1}{\sin \theta }$
$=\frac{1-{\sin }^2\theta }{\sin \theta };\frac{dx}{d\theta }=\frac{{\cos }^2\theta }{\sin \theta };\frac{dy}{d\theta }=\cos \theta$
$\frac{dy}{dx}=0;\tan \theta =0$
$\theta =n\pi ,n\in z$