Can't factor any further. I am taking Pre-Cal over the summer online but sometimes, that is a bit detrimental because of the lack of a teacher. I am supposed to prove $(\sin x)(\tan x \cos x - \cot x \cos x) = 1 - 2 \cos^2 x$. So far, I have gotten to $\sin^2 x - \cos^2 x = 1- 2\cos^2 x$ but I don't know what else I can do.

If you've transformed the left hand side into $\sin^2 x - \cos^2 x$, then you're nearly there! Just apply the Pythagorean Identity $(\sin^2 \theta + \cos^2 \theta = 1)$ so that:

$$ \sin^2 x - \cos^2 x = (1 - \cos^2x) - \cos^2 x = 1 - 2\cos^2 x $$