Question

If y = sq.root [{1-cos x}/{1+ cos x}], find dy / dx.

Answer 1

$$\begin{gathered} y=\sqrt{\frac{1-\cos x}{1+\cos x}} \\ \Rightarrow y=\sqrt{\left(\frac{1-\cos x}{1+\cos x}\right)\times \left(\frac{1-\cos x}{1-\cos x}\right)} \\ \Rightarrow y=\sqrt{\frac{{\left(1-\cos x\right)}^2}{1-{\cos }^{2}x}} \\ \Rightarrow y=\sqrt{\frac{{\left(1-\cos x\right)}^2}{{\sin }^{2}x}} \\ \Rightarrow y=\frac{1-\cos x}{\sin x} \\ \Rightarrow y=\mathrm{cosec}x-\cot x \\ \Rightarrow \frac{dy}{dx}=-\cot x\mathrm{cosec}x+{\mathrm{cosec}}^{2}x \end{gathered}$$