Question
Diffrentiate the following w.r.t.x
$\log \left[\tan \left(\frac{x}{2}\right)\right]$
$\log \left[\tan \left(\frac{x}{2}\right)\right]$
Differentiating w.r.t. x, we get
$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x} \log \left[\tan \left(\frac{x}{2}\right)\right] \\ & =\frac{1}{\tan \left(\frac{x}{2}\right)} \cdot \frac{d}{d x}\left[\tan \left(\frac{x}{2}\right)\right] \\ & =\frac{1}{\tan \left(\frac{x}{2}\right)} \cdot \sec ^2\left(\frac{x}{2}\right) \cdot \frac{d}{d x}\left(\frac{x}{2}\right)\end{aligned}$
$\begin{aligned} & =\frac{\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)} \cdot \frac{1}{\cos ^2\left(\frac{x}{2}\right)} \cdot \frac{1}{2} \times 1 \\ & =\frac{1}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)} \\ & =\frac{1}{\sin x}=\operatorname{cosec} x .\end{aligned}$
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