Question
If $x=\ln \tan \frac{\theta}{2}$ and $y=\tan \theta-\theta$, prove that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\tan ^{2} \theta \sin \theta(\cos \theta+2 \sec \theta)$
Step 1
\begin{align*} \frac{dx}{d\theta} &= \frac{d}{d\theta} \ln \tan \frac{\theta}{2} \\ &= \frac{1}{\tan \frac{\theta}{2}} \cdot \frac{d}{d\theta} \tan \frac{\theta}{2} \\ &= \frac{1}{\tan \frac{\theta}{2}} \cdot \sec^2 \frac{\theta}{2} \\ &= \frac{\sec^2 Show more…
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