$\begin{align}  & \text{From the left hand side} \\ & \sqrt{\frac{1-\sin x}{1+\sin x}}=\sqrt{\frac{1-\sin x}{1+\sin x}\times \frac{1+\sin x}{1+\sin x}} \\ & \sqrt{\frac{1-\sin x}{1+\sin x}}=\sqrt{\frac{1-{{\sin }^{2}}x}{{{(1+\sin x)}^{2}}}}=\sqrt{\frac{{{\cos }^{2}}x}{{{(1+\sin x)}^{2}}}} \\ & \sqrt{\frac{1-\sin x}{1+\sin x}}=\frac{\left| \cos x \right|}{1+\sin x} \\\end{align}$