Ask Me Help Desk - How do you prove this identity?

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How do you prove this identity?

(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? :)

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