If \(x=a (\cos t + \log \tan\displaystyle\frac{t}{2}), y= a \sin t\), then value of \(\displaystyle\frac{dy}{dx}\) at \(t=\displaystyle\frac{\pi}{4}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
For a curve given in terms of a parameter t,
\( \displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \)
Here
\( \displaystyle \frac{dy}{dt} = a\cos t \)
For the log term, by the chain rule:
\( \displaystyle \frac{d}{dt}\log\tan\frac{t}{2} = \frac{1}{\tan\frac{t}{2}}\cdot\sec^2\frac{t}{2}\cdot\frac{1}{2} \)
\( \displaystyle = \frac{1}{2\sin\frac{t}{2}\cos\frac{t}{2}} = \frac{1}{\sin t} \)
So
\( \displaystyle \frac{dx}{dt} = a\left(-\sin t + \frac{1}{\sin t}\right) \)
\( \displaystyle = \frac{a(1 - \sin^2 t)}{\sin t} = \frac{a\cos^2 t}{\sin t} \)
Then
\( \displaystyle \frac{dy}{dx} = a\cos t\cdot\frac{\sin t}{a\cos^2 t} = \tan t \)
At \(t = \frac{\pi}{4}\):
\( \displaystyle \frac{dy}{dx} = \tan\frac{\pi}{4} = 1 \)