To solve the equation cot(x + 2π/3) = cot(π/4), we can use the identity cot(x) = 1/tan(x).

So, cot(x + 2π/3) = cot(π/4) can be rewritten as 1/tan(x + 2π/3) = 1/tan(π/4).

Since the cotangent function is equal to the reciprocal of the tangent function, we can simplify the equation to tan(x + 2π/3) = tan(π/4).

Now, we can use the periodicity of the tangent function to find the solutions.

The tangent function has a period of π, which means that tan(x + π) = tan(x) for any value of x.

So, tan(x + 2π/3) = tan(x + 2π/3 + π) = tan(x + 8π/3).

Therefore, we can rewrite the equation as tan(x + 8π/3) = tan(π/4).

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

Using the identity tan(π/4) = 1, we have tan(x + 8π/3) = 1.

Since the tangent function has a period of π, we can find the solutions by setting x + 8π/3 = π/4 + nπ, where n is an integer.

Solving for x, we have x = π/4 + nπ - 8π/3.

Therefore, the solutions to the equation cot(x + 2π/3) = cot(π/4) are x = π/4 + nπ - 8π/3, where n is an integer.