To solve the equation, we can use the double angle formula for cosine:

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

Substituting this into the equation, we have:

2cos^2(x) - 1 - cos(x) = 0

Rearranging the terms, we get:

2cos^2(x) - cos(x) - 1 = 0

Let's solve this quadratic equation for cos(x). We can factor it as:

(2cos(x) + 1)(cos(x) - 1) = 0

Setting each factor equal to zero, we have:

2cos(x) + 1 = 0 or cos(x) - 1 = 0

Solving the first equation, we get:

2cos(x) = -1

cos(x) = -1/2

Using the unit circle or a calculator, we find that the solutions for cos(x) = -1/2 are:

x = 2pi/3 + 2npi or x = 4pi/3 + 2npi

where n is an integer.

Now, let's solve the second equation:

cos(x) - 1 = 0

cos(x) = 1

Using the unit circle or a calculator, we find that the solution for cos(x) = 1 is:

x = 2npi

where n is an integer.

Therefore, the solutions to the equation cos(2x - pi/3) = -cos(x) are:

x = 2pi/3 + 2npi, 4pi/3 + 2npi, or 2npi, where n is an integer.