How do you prove 1-4sin^4x=cos^2x(1+2sin^2x)?

1 Answer
May 29, 2018

I don't think 1-4sin^4x=color(blue)(cos^2x)(1+2sin^2x) is valid for all x.

I got 1-4sin^4x=color(blue)(cos2x)(1+2sin^2x)

Explanation:

Here's my proof

1-4sin^4x
=1-(2sin^2x)^2
=1-(1-cos2x)^2 (cosine double angle formula)
=1-(1-2cos2x+cos^2(2x)) (expanding)
=2cos2x-cos^2(2x) (expanding)
=cos2x(2-cos2x) (factorising)
=cos2x(1+1-cos2x) (1+1=2)
=cos2x(1+2sin^2(x)) (cosine double angle formula)