If ${\displaystyle \int \frac{Sinx}{Cosx(1+\cos x)}}dx=f\left(x\right)+c$ $\Rightarrow$ $f\left(x\right)=$

          $\cos x=t\Rightarrow -\sin xdx=dt$

             ${\displaystyle \int \frac{-1}{t\left(1+t\right)}}dt={\displaystyle \int \left(\frac{1}{t+1}-\frac{1}{t}\right)}dt$

                                   $=\log \left|\frac{1+t}{t}\right|+c$

                                   $=\log \left|\frac{1+\cos x}{\cos x}\right|+c$