To solve this equation, let's first apply the double-angle formula for cosine, which states that:

cos(2x) = 2cos^2(x) - 1

Now we can rewrite the equation as:

2(2cos^2(x) - 1) = 7cos(x)

Expanding, we get:

4cos^2(x) - 2 = 7cos(x)

Rearranging the terms, we get a quadratic equation in terms of cosine:

4cos^2(x) - 7cos(x) - 2 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula:

cos(x) = [7 ± sqrt(7^2 - 4(4)(-2))] / (2(4))

cos(x) = [7 ± sqrt(49 + 32)] / 8

cos(x) = [7 ± sqrt(81)] / 8

cos(x) = (7 ± 9) / 8

This gives us two possible solutions for cos(x) which are:

cos(x) = 1 or cos(x) = -1/4

Since the cosine function is periodic (repeats every 2π), the general solution will be:

x = 2πn or x = 2πn ± arccos(-1/4)

where n is an integer.