Use the power reducing formulas to rewrite cos^4(4x) without exponents

Answer 1

The power reducing formula for cosine is cos²(x) = (1+cos(2x))/2.

With that in mind, cos⁴(4x) = cos²(4x) * cos²(4x)

= (1+cos(8x))/2 * (1+cos(8x))/2 = 1/2 * (1+cos(8x))²

= 1/2 * (1 + 2cos(8x) + cos²(8x))

= 1/2 * (1 + 2cos(8x) + (1+cos(16x))/2)

= 1/2 * (1 + 2cos(8x) + 1/2 + cos(16x)/2)

= 1/2 + cos(8x) + 1/4 + cos(16x)/4

= (3 + 4cos(8x) + cos(16x))/8