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

1 Answer
e52fa787
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)

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