We have to prove that \(\displaystyle{\left({1}-\frac{{{{\cos}^{{2}}{\left({x}\right)}}}}{{{1}+{\sin{{\left({x}\right)}}}}}\right)}={\sin{{\left({x}\right)}}}\)

Let us start from the left hand side. Note that \(\displaystyle{{\sin}^{{2}}{x}}+{{\cos}^{{2}}{x}}={1}\). Then

\((1-(\cos^2(x))=\frac{1+\sin(x)-\cos^2(x)}{1+(\sin(x))} =(\sin(x))+[1-\cos^2(x)]\)

\(=\frac{\sin(x)+\sin^2(x)}{1+(\sin(x))} =\frac{\sin(x)(1+\sin(x))}{1+\sin(x)} =\sin(x)\)

Hence the proof.