Answer:
(3/8) + (1/8)cos 4x - (1/2)cos 2x
Step-by-step 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
