(a)OR(b)

Express $\displaystyle \tan ^{-1}\left(\frac{\cos x}{1-\sin x}\right)$, where $\displaystyle \frac{-\pi}{2}<x<\frac{\pi}{2}$ in the simplest form.

$$\begin{aligned} & y=\tan ^{-1}\left[\frac{\cos x}{1-\sin x}\right]=\tan ^{-1}\left[\frac{\cos ^{2} \dfrac{x}{2}-\sin ^{2} \dfrac{x}{2}}{\left(\cos \dfrac{x}{2}-\sin \dfrac{x}{2}\right)^{2}}\right] \\ & y=\tan ^{-1}\left[\frac{1+\tan \dfrac{x}{2}}{1-\tan \dfrac{x}{2}}\right]=\tan ^{-1}\left[\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right]=\left(\frac{\pi}{4}+\frac{x}{2}\right) \end{aligned} \begin{aligned} & \tan ^{-1}(1)+\left[\pi-\cos ^{-1}\left(\frac{1}{2}\right)\right]-\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}+\left(\pi-\frac{\pi}{3}\right)-\frac{\pi}{4} \\ & =\frac{2 \pi}{3} \end{aligned} $$