To solve this equation, we can use the identity:

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

So, the equation becomes:

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

Adding cos(x) to both sides:

1 - 2sin^2(x) = 0

Rearranging the equation:

2sin^2(x) = 1

Dividing both sides by 2:

sin^2(x) = 1/2

Taking the square root of both sides:

sin(x) = ±√(1/2)

Now, we need to find the values of x that satisfy this equation.

sin(x) = √(1/2) when x = π/4 or x = 3π/4

sin(x) = -√(1/2) when x = 5π/4 or x = 7π/4

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

x = π/4, 3π/4, 5π/4, 7π/4