How do you prove #cos^4(x)-sin^4(x)=cos2x#?

1 Answer
Kevin B.
Apr 19, 2015

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

Remember the double angle formula for cosine:
#color(blue)(cos(2x)=cos^2(x)-sin^2(x)#

Plugging it into the right hand side:
#cos^4(x) - sin^4(x) = cos^2(x)-sin^2(x)#

Using differences of squares on the left side:
#(cos^2(x) + sin^2(x)) (cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)#

And since #color(blue)(cos^2(x) + sin^2(x)=1#
#1(cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)#

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