Question

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

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

Answer 1

The correct option is A

$\tan t$

Explanation for the correct answer:

To find the value of $\frac{dy}{dx}$ :

Given,

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

Now, Differentiate $x$ and $y$ with respect to $t$.

$\begin{array}{rcl}\frac{dy}{dt}& =& a\cos t\\ x& =& a(\cos t+\log \tan \frac{t}{2})\\ \frac{dx}{dt}& =& a\left(-\sin t+\left(\frac{1}{\tan \frac{t}{2}}\right)se{c}^2\left(\frac{t}{2}\right)\left(\frac{1}{2}\right)\right)\\ & =& a\left(-\sin t+\left(\frac{1}{\sin t}\right)\right)\\ & =& \frac{a(1–{\sin }^{2}t)}{\sin t}\\ & =& a\frac{{\cos }^{2}t}{\sin t}\\ \frac{dy}{dx}& =& \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}\\ & =& \frac{a\cos t}{a{\displaystyle \frac{{\cos }^{2}t}{\sin t}}}\\ & =& \tan t\end{array}$

Hence, the correct option is A.