1+cosx=(sin^2(x))/(1-cosx)

\frac{sin^{2}(x)}{1-cos(x)}

Multiply top and bottom by
1+cos(x):

\frac{sin^{2}(x)}{(1-cos(x))}\cdot \frac{(1+cos(x))}{(1+cos(x))}

The denominator becomes

1-cos^{2}(x)=sin^{2}(x)

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

Notice that the sin^{2}(x) cancels and we are left with

1+cos(x)

QED