To prove the given equation, we will start by using the trigonometric identity:

sin(2x) = 2sin(x)cos(x)

Now, we substitute this identity into the given equation:

sin(x) + 2sin(x)cos(x) = cos(x) + 2cos^2(x)

Rearranging terms, we get:

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

Now, we will use the Pythagorean identity, sin^2(x) + cos^2(x) = 1, to express sin^2(x) in terms of cos(x):

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

Substitute sin^2(x) = 1 - cos^2(x) into the equation we derived earlier:

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

Now, simplify the right-hand side of the equation:

sin(x) = cos(x) + 2cos^2(x) - 2cos(x) + 2cos^3(x)

Combine like terms:

sin(x) = -cos(x) + 2cos^2(x) + 2cos^3(x)

Now, we can see that the equation we derived does not match the original equation given. Therefore, the given equation sin(x) + sin(2x) = cos(x) + 2cos^2(x) is not correct.