Question

If $x=a\left(\cos \theta +\log \tan \frac{\theta }{2}\right)$ and $y=\sin \theta$ then find the value of $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{4}$

Answer 1

Given, $x=a(\cos \theta +\log \tan \frac{\theta }{2})$

$\frac{d^{2}x}{d{\theta }^2}=a(-\cos \theta -\cot \theta \csc \theta )$

$y=\sin \theta$

$\frac{d^{2}y}{d{\theta }^2}=-\sin \theta$

$\frac{d^{2}y}{d{x}^2}=\frac{-\sin \theta }{a(-\cos \theta -\cot \theta cosec\theta )}$

$=\frac{\sin \theta }{a(\cos \theta +\cot \theta cosec\theta )}$

$=\frac{\sin \left(\frac{\pi }{4}\right)}{a(\cos \left(\frac{\pi }{4}\right)+\cot \left(\frac{\pi }{4}\right)\csc \left(\frac{\pi }{4}\right))}$

$=\frac{\frac{\sqrt{2}}{2}}{(\frac{\sqrt{2}}{2}+\sqrt{2})a}$

$=\frac{1}{3a}$