$$\begin{gathered} \frac{dy}{dx}=\frac{-\cos x}{1+\sin x} \\ \end{gathered}$$

$$\begin{gathered} \Rightarrow dy=\left(\frac{-\cos x}{1+\sin x}\right)dx \\ \end{gathered}$$

$$\begin{gathered} y=\ln \left|\frac{1}{1+\sin x}\right|+c \\ \end{gathered}$$

$$\begin{gathered} Since,f\left(0\right)=0 \\ \end{gathered}$$

$$\begin{gathered} \Rightarrow 0=0+c\Rightarrow c=0 \\ \end{gathered}$$

$$\begin{gathered} f\left(x\right)=\ln \left|\frac{1}{1+\sin x}\right| \\ \end{gathered}$$

f\left(\frac{\pi }{2}\right)=\ln \left(\frac{1}{2}\right)=-\ln 2