Final answer:
After applying trigonometric identities, it is shown that sin 3x = 3 sin x - 4 sin^3 x, which is the triple angle formula for sine. This result does not match the given expression 4 cos^3 x - 3 cos x, indicating that the original statement is not an identity.
Step-by-step explanation:
To verify whether sin 3x = 4 cos3 x − 3 cos x is an identity, we can use known trigonometric identities to transform one side of the equation and see if it matches the other side. The identity in question is known as the triple angle formula for sine, which says that sin 3x can be expressed in terms of powers of sine and cosine for the angle x.
Let's start with the right-hand side of the equation, 4 cos3 x − 3 cos x and apply the following identities:
- cos 2x = 2 cos2 x - 1
- cos2 x = 1 - sin2 x
We know that sin 2x = 2 sin x cos x and this can be extended to the triple angle identity:
sin 3x = sin (2x + x) = sin 2x cos x + cos 2x sin x = 2 sin x cos2 x + (2 cos2 x - 1) sin x).
By substituting cos2 x = 1 - sin2 x, we arrive at the expression:
sin 3x = 2 sin x (1 - sin2 x) + (2(1 - sin2 x) - 1) sin x = 2 sin x - 2 sin3 x + 2 sin x - 2 sin3 x - sin x.
Which simplifies to sin 3x = 3 sin x - 4 sin3 x. This is the known triple angle formula for sine. However, this is not the same as the expression given (4 cos3 x - 3 cos x), implying that the original statement is not an identity.