I'm beyond lost here, is there a way to prove that?

sec^{2}(x)=\frac{csc(x)}{csc(x)-sin(x)}

Start with the right hand side:

Since csc(x)=\frac{1}{sin(x)}:

\frac{\frac{1}{sin(x)}}{\frac{1}{sin(x)}-sin(x)}

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

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

sin(x) cancels and we are left with:

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

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

\frac{1}{cos^{2}(x)}=sec^{2}(x)

Thank you! You're brilliant!