Here are two that I cannot seem to prove. Any and all help is appreciated.
$\displaystyle \frac{\cos^2(x) - \sin^2(x)}{1-\tan^2(x)} = \cos^2(x)$
$\displaystyle \sin(2t) - \tan(t) = \tan(t)\cos(2t)$
start with the LHS. write $\displaystyle \tan^2 x$ as $\displaystyle \frac {\sin^2 x}{\cos^2 x}$ and combine the fractions in the denominator. you should see the answer almost immediately.
start with the LHS. write $\displaystyle \sin 2t$ as $\displaystyle 2 \sin t \cos t$ and $\displaystyle \tan t$ as $\displaystyle \frac {\sin t}{\cos t}$. simplify. you should see what to do from there$\displaystyle \sin(2t) - \tan(t) = \tan(t)\cos(2t)$
as you are seeing, it is very important to know your standard trig identities for these problems