Recall that:
cos^2(x) = (1 + cos(2x))/2
cos(2x) = 2cos^2(x) - 1
Using these identities, we can rewrite cos^4(x) as follows:
cos^4(x) = (cos^2(x))^2
= [(1 + cos(2x))/2]^2
= (1/4) + (1/2)cos(2x) + (1/4)cos^2(2x)
= (1/4) + (1/2)cos(2x) + (1/4)[2cos^2(x) - 1]^2
= (1/4) + (1/2)cos(2x) + (1/4)[4cos^4(x) - 4cos^2(x) + 1]
Now, we can simplify the expression by collecting like terms and grouping the powers of cos(x) together:
cos^4(x) = (1/4) + (1/2)cos(2x) + (1/4)[4cos^4(x) - 4cos^2(x) + 1]
= (3/4)cos^4(x) - (1/2)cos^2(x) + (5/4)
= (3/4)[1 - sin^2(x)]^2 - (1/2)[1 - sin^2(x)] + (5/4)
This final expression is equivalent to cos^4(x) and does not contain any powers of trigonometric functions greater than 1.