(tan^4x)-(tan^2x)(sec^2x) = -tan^2x

You can do it in a very simple way starting by factoring tan^2x.

\tan^4x - \tan^2x \sec^2x = \tan^2x \tan^2x - \tan^2x \sec^2x

=\tan^2x(\tan^2x - \sec^2x)

Then, recall the pythagoras' identify:

\cos^2x + \sin^2x = 1

Divide this by cos^2x to give:

1 + \tan^2x = \sec^2x

This means that \tan^2x - \sec^2x = -1

Substitute that in

=\tan^2x(\tan^2x - \sec^2x)

What do you get? :)