Question
Prove the identity:
cos^4 x + sin^4 x = 1 - (1/2)sin^2 (2x)
cos^4 x + sin^4 x = 1 - (1/2)sin^2 (2x)
Answers
Reiny
LS = cos^4 x + sin^4 x
= (cos^2 x + sin^2 x)^2 - (2sin^2 x)(cos^2 x)
= 1 - 2(sin^2 x)(cos^2 x) , now recall sin(2x) = 2sinxcosx
= 1 - (2sinxcosx)(sinxcosx)
= 1 - sin(2x) (1/2)sin(2x)
= 1 - (1/2) sin^2 (x)
= RS
= (cos^2 x + sin^2 x)^2 - (2sin^2 x)(cos^2 x)
= 1 - 2(sin^2 x)(cos^2 x) , now recall sin(2x) = 2sinxcosx
= 1 - (2sinxcosx)(sinxcosx)
= 1 - sin(2x) (1/2)sin(2x)
= 1 - (1/2) sin^2 (x)
= RS