rewrite the expression in terms of the first power of the cosine.
sin^8x
To rewrite the expression sin^8x in terms of the first power of cosine, we can use the trigonometric identity:
sin^2x = 1 - cos^2x
Using this identity, we can rewrite the expression as:
(sin^2x)^4
Now, substitute sin^2x with 1 - cos^2x:
(1 - cos^2x)^4
Expanding this expression using the binomial theorem, we get:
1 - 4cos^2x + 6cos^4x - 4cos^6x + cos^8x
Therefore, sin^8x can be rewritten in terms of the first power of cosine as:
1 - 4cos^2x + 6cos^4x - 4cos^6x + cos^8x
Recall that sin^2(x) = 1/2 (1-cosx)
use that several times to end up with
sin^4x = 1/8 (cos4x - 4cos2x + 3)
and on to
1/256 (cos8x - 8cos6x +28cos4x - 56cos2x + 35