cos^2(x)(1+6cos(x)) = 1
cos^2(x) + 6cos^3(x) = 1
cos^2(x) = 1 - 6cos^3(x)

Substitute cos^2(x) = 1 - sin^2(x):

1 - sin^2(x) = 1 - 6cos^3(x)
sin^2(x) = 6cos^3(x)

Then we know that sin^2(x) = 1 - cos^2(x), so we can substitute the equation:

1 - cos^2(x) = 6cos^3(x)
cos^2(x) + 6cos^3(x) - 1 = 0

This is a cubic equation in terms of cos(x), and can be solved using various methods such as synthetic division or numerical methods.