Verify the following identity:
3/8 - (cos(2 x))/(2) + (cos(4 x))/(8) = sin(x)^4
Put 3/8 - 1/2 cos(2 x) + 1/8 cos(4 x) over the common denominator 8: 3/8 - 1/2 cos(2 x) + 1/8 cos(4 x) = (3 - 4 cos(2 x) + cos(4 x))/8:
(3 - 4 cos(2 x) + cos(4 x))/8 = ^?sin(x)^4
Multiply both sides by 8:
3 - 4 cos(2 x) + cos(4 x) = ^?8 sin(x)^4
cos(2 x) = 1 - 2 sin(x)^2:
3 - 41 - 2 sin(x)^2 + cos(4 x) = ^?8 sin(x)^4
-4 (1 - 2 sin(x)^2) = 8 sin(x)^2 - 4:
3 + 8 sin(x)^2 - 4 + cos(4 x) = ^?8 sin(x)^4
cos(4 x) = 1 - 2 sin(2 x)^2:
3 - 4 + 8 sin(x)^2 + 1 - 2 sin(2 x)^2 = ^?8 sin(x)^4
sin(2 x) = 2 sin(x) cos(x):
3 - 4 + 8 sin(x)^2 + 1 - 2 2 cos(x) sin(x)^2 = ^?8 sin(x)^4
Multiply each exponent in 2 sin(x) cos(x) by 2:
3 - 4 + 8 sin(x)^2 + 1 - 24 cos(x)^2 sin(x)^2 = ^?8 sin(x)^4
cos(x)^2 = 1 - sin(x)^2:
3 - 4 + 8 sin(x)^2 + 1 - 2×4 1 - sin(x)^2 sin(x)^2 = ^?8 sin(x)^4
4 (1 - sin(x)^2) sin(x)^2 = 4 sin(x)^2 - 4 sin(x)^4:
3 - 4 + 8 sin(x)^2 + 1 - 24 sin(x)^2 - 4 sin(x)^4 = ^?8 sin(x)^4
3 - 4 + 8 sin(x)^2 + 1 - 2 (4 sin(x)^2 - 4 sin(x)^4) = 8 sin(x)^4:
8 sin(x)^4 = ^?8 sin(x)^4
The left hand side and right hand side are identical:
Answer: |(identity has been verified)
