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)) = 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(xcos(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.
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)) =
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.