Question

Rewrite sin^{4}x so that it involves only the first power of cosine

Show work and reasoning. Please helpp

Answer 1

(3/8) + (1/8)cos 4x - (1/2)cos 2x

Explanation:

sin^(4) x = (sin² x ) (sin² x) ------ (1)

Recall double angle identity: cos 2x = 1 - 2 sin² x

or sin² x = (1/2) (1 - cos 2x)

substituting this into (1)

sin^(4) x = (sin² x ) (sin² x) = [ (1/2)(1-cos 2x) ]²

= (1/4) ( 1 - 2 cos 2x + cos²2x) ------(2)

Also recall double angle identiy : cos 2x = 2 cos² x - 1

or cos² x = (1/2) ( 1+ cos 2x)

or cos² 2x = (1/2) ( 1+ cos 4x)

substituting this into (2)

(1/4) ( 1 - 2 cos 2x + cos²2x)

= (1/4) [ 1 - 2 cos 2x + (1/2)(1 + cos 4x) ] (expand and reduce)

= (3/8) + (1/8)cos 4x - (1/2)cos 2x