#sin^4x-cos^4x=1-2cos^2x# prove it?
2 Answers
Apr 27, 2018
We want to show that
We'll work with the LHS:
Using the identity
Apr 27, 2018
See explanation...
Explanation:
We will use Pythagoras' identity:
#sin^2 x + cos^2 x = 1#
from which we can deduce:
#sin^2 x = 1 - cos^2 x#
Also note that the difference of squares identity can be written:
#A^2-B^2 = (A-B)#
We can use this with
#sin^4 x - cos^4 x = (sin^2 x)^2 - (cos^2 x)^2#
#color(white)(sin^4 x - cos^4 x) = (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)#
#color(white)(sin^4 x - cos^4 x) = sin^2 x - cos^2 x#
#color(white)(sin^4 x - cos^4 x) = (1-cos^2 x) - cos^2 x#
#color(white)(sin^4 x - cos^4 x) = 1-2cos^2 x#