Rewrite cos 4x in terms of cos x.

Answer 1

So in this case, we rewrite cos4x into the addition formula identity

cos(4x) = cos(2x +2x)

cos(2x+2x) = cos(2x)cos(2x)-sin(2x)sin(2x)

we then use our trigonometric identities with the double angle formula to tell us what cos(2x) and sin(2x) is

cos(4x) = (cos²(x)-sin²(x))*(cos²(x)-sin²(x)) - (2sin(x)cos(x))*(2sin(x)cos(x))

we know from pythagorean's theorem that sin²(x) = 1-cos²(x)

so we substitute our sin²(x) first

cos(4x) = (cos²(x)-1+cos²(x))*(cos²(x)-1+cos²(x)) - 4sin²(x)cos²(x)

do the same substitution again and simplify even more

cos(4x) = (2cos²(x)-1)*(2cos²(x)-1) - 4(1-cos²(x))*cos²(x)

cos(4x) = 4cos⁴(x) - 4cos²(x) + 1 - 4cos²(x) + 4cos⁴(x)

cos(4x) = 8cos⁴(x) - 8cos²(x) + 1

Let me know if you have any questions on the steps.