How do you factor the expression and use the fundamental identities to simplify #1-2cos^2x+cos^4x#?

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Answer 1 · 2017-10-31T10:43:14

The answer is #=sin^4x#

We need

#(a-b)^2=a^2-2ab+b^2#

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

Therefore,

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

#=1(1-cos^2x)-cos^2x(1-cos^2x)#

#=(1-cos^2x)(1-cos^2x)#

#=(1-cos^2x)^2#

#=sin^4x#

Answer 2 · 2017-10-31T14:02:50

#color(blue)(sin^4x)#

This makes use of the Pythagorean identity:

#sin^2x + sin^2x = 1#

#1-2cos^2x+cos^4x#

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

#cos^2x= 1-sin^2x#

#:.#

#1 -2(1-sin^2x)+1-2sin^2x+sin^4x#

#1 -2+2sin^2x+1-2sin^2x+sin^4x#

#1 -2+2sin^2x+1-2sin^2x+sin^4x=color(blue)(sin^4x)#