To solve this equation, we will first use the identity that relates sine and cosine:

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

Now, let's rewrite the given equation using this identity:

sin(x) + sin(5x) - 2(1 - 2sin^2(x)) = 0
sin(x) + sin(5x) - 2 + 4sin^2(x) = 0

Next, let's use the angle addition formula for sine:

sin(5x) = sin(3x + 2x) = sin(3x)cos(2x) + cos(3x)sin(2x)

Now, substitute sin(5x) with the angle addition formula:

sin(x) + sin(3x)cos(2x) + cos(3x)sin(2x) - 2 + 4sin^2(x) = 0

Now, we have a mixed trigonometric equation that involves both sine and cosine terms. We can simplify further by expanding the sine and cosine terms using the trigonometric identities for sin(2x) and cos(2x):

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

By substituting these into our equation, we will have a polynomial equation in terms of sin(x) and cos(x) that we can solve using algebraic methods.