15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T66a,

$$\begin{aligned} & \text { } z=\log \left(\tan \left(\frac{x}{2}\right)\right) \\ & \Rightarrow \frac{d z}{d x}=\frac{1}{\tan \left(\frac{x}{2}\right)} \times \sec ^2\left(\frac{x}{2}\right) \times \frac{1}{2} \end{aligned}$$

$$=\frac{1}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}=\frac{1}{\sin x}=\operatorname{cosec} x$$ ..... (i)

$$\therefore \quad \frac{d y}{d x}=\frac{d y}{d z} \times \frac{d z}{d x}$$ ..... (ii)

$$=\operatorname{cosec} x \frac{d y}{d z}$$

$$\begin{aligned} \frac{d^2 y}{d x^2} & =\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d x}\left(\operatorname{cosec} x \frac{d y}{d z}\right) \\ & =\frac{d}{d z}\left(\operatorname{cosec} x \frac{d y}{d z}\right) \times \frac{d z}{d x} \end{aligned}$$

$$\begin{aligned} & {\left[\operatorname{cosec} x \frac{d^2 y}{d z^2}+\frac{d y}{d z} \times(-\operatorname{cosec} x \cot x)\left(\frac{d x}{d z}\right)\right]\left(\frac{d z}{d x}\right)} \\ & =\left[\operatorname{cosec} x \frac{d^2 y}{d z^2}-\frac{d y}{d z} \times \operatorname{cosec} x \cot x \times \frac{1}{\operatorname{cosec} x}\right] \operatorname{cosec} x \\ & =\operatorname{cosec}^2 x \frac{d^2 y}{d z^2}-\operatorname{cosec} x \cot x \frac{d y}{d z} \\ & \therefore \quad \frac{d^2 y}{d x^2}+\cot x \frac{d y}{d x}+4 y \operatorname{cosec}^2 x=0 \end{aligned}$$

$$\begin{aligned} \Rightarrow & \quad\operatorname{cosec}^2 x \frac{d^2 y}{d z^2}-\operatorname{cosec} x \cot x \frac{d y}{d z} \\ & +\operatorname{cosec} x \cot x \frac{d y}{d z}+4 y

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