Question

If x=a$cos\Theta$+a log tan $\frac{\Theta }{2}$ and y= a$sin\Theta$, then $\frac{dy}{dx}$ is equal to

• A. $cot\Theta$
• B. $tan\Theta$
• C. $sin\Theta$
• D. $cos\Theta$

Answer 1

The correct option is B $tan\Theta$

$x=a\cos \theta +alog\tan \frac{\theta }{2}$ $y=a\sin \theta$

$\frac{dx}{d\theta }=a(-\sin \theta +\frac{1}{2}\frac{{\sec }^2\theta /2}{\tan \theta /2})$ $\frac{dy}{d\theta }=a\cos \theta$

$=a(\frac{1}{2\sin \theta /2\cos \theta /2}-\sin \theta )$

$=a(\frac{1}{2\sin \theta }-\sin \theta )$

$=\frac{a{\cos }^2\theta }{\sin \theta }$

$\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$

$=\frac{a\cos \theta }{a\cos \theta }\times \sin \theta$

$=\frac{\sin \theta }{\cos \theta }$

$=\tan \theta$.