Please solve this problem!

Question

Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.

cos⁴(3x)

Answer 1

I will show two ways.

Method 1.

As cos^2(a) = 1/2 + 1/2 cos 2a,

cos^4(3x) =

(cos^2(3x))^2 =

(1/2 + 1/2 cos(6x))^2 =

1/4 + 1/2 cos(6x) + 1/4 cos^2(6x) =

1/4 + 1/2 cos(6x) + 1/8 cos(12x) + 1/8 =

3/8 + 1/2 cos(6x) + 1/8 cos(12x)

Method 2: as cos(a) = 1/2(e^ia + e^-ia),

cos^4(3x) =

[(e^3ix + e^-3ix)/2]^4 =

e^12ix/16 + 4e^6ix/16 + 6/16 + 4e^-6ix/16 + e^-12ix =

1/16(e^12ix+e^-12ix) + 4/16(e^6ix/16+e^-6ix) + 6/16

1/8 cos(12x) + 1/2 cos(6x) + 3/8