#1**+1 **

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)**

Guest Mar 18, 2017