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Q)

If $ x=\cos \theta + log \tan \Large\frac{\theta }{2},\normalsize y=\sin \theta \;find\; \Large\frac {d^2y}{dx^2}\normalsize \;at\;\theta =\Large\frac{\pi}{4}$

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A)
Toolbox:
  • $\large \frac{dy}{dx}=\large\frac{\large\frac{dy}{d\theta}}{\large\frac{dx}{d\theta}}$
  • $ 2sin\large\frac{\theta}{2}cos\large\frac{\theta}{2}=sin\theta$
  • $\large \frac{d^2y}{dx^2}=\large\frac{d}{d\theta} \bigg( \large\frac{dy}{dx} \bigg) $ x $\large \frac{d\theta}{dx} $
$ \large\frac{dx}{d\theta} = -sin\theta + \large\frac{1}{tan\large\frac{\theta}{2}}$ x $ sec^2\large\frac{\theta}{2}$ x $ \large\frac{1}{2}$
$ = -sin\theta + \large\frac{1}{sin\theta}=\large\frac{cos^2\theta}{sin\theta}=cosec\theta cot\theta$
$ \large\frac{dy}{d\theta}=cos\theta$
$\large\frac{dy}{dx}=\large\frac{cos\theta}{cos^2\theta}$x$sin\theta=tan\theta$
$\large \frac{d^2y}{dx^2}=sec^2\theta$x$\large\large\frac{sin\theta}{cos^2\theta}=\large\frac{sin\theta}{cos^4\theta}$
$ \large\frac{d^2y}{dx^2}\: when \: \theta=\large\frac{\pi}{4}\: is\: \large\frac{4}{\sqrt 2}=2\sqrt2$

 

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