Question

(1-cos^2x)(1+cos^2x)=2sin^2x-sin^4x

Answer 1

You have to prove the identity \((1-\cos^2x)(1+\cos^2x)=2\sin^2x-\sin^4x\).

Maybe you have already tried multiplying out the LHS.
This is tempting, because it has the form \((a-b)(a+b)\), which is equal to \(a^2-b^2\).
Here, it would be \(1-\cos^4x\).

This is not so nice, because now you have a fourth power of cos, while you only need sines.

This will work: remember that famous identity: \(\sin^2x+\cos^2x=1\)?
It can be used to convert from cos to sin:

\(\cos^2x=1-\sin^2x\).
Put this in place of the \(\cos^2x\) in the LHS, and you will arrive quickly at the RHS!

Answer 2

Just try it. If you don't succeed, I can help you with it!