Final answer:
The expression (sin 3x)(cos x) - (cos 3x)(sin x) simplifies to sin 2x.
Step-by-step explanation:
The expression (sin 3x)(cos x) - (cos 3x)(sin x) simplifies to sin(3x - x). Using the trigonometric identity sin(a - b) = sin a cos b - cos a sin b, we can simplify further to sin 2x.
The expression (sin 3x)(cos x) − (cos 3x)(sin x) simplifies to sin(3x - x), which is sin(2x). This simplification is possible due to the sum-to-product trigonometric identities, which allow us to simplify sums and differences of sinusoidal functions. Specifically, the identity we use here is sin u − sin v = 2 sin((u-v)/2) cos((u+v)/2), where in our case, u = 3x and v = x.
What are trigonometric ratios?
Trigonometric ratios are ratios of the sides of a right triangle that are used to define the relationships between the angles and the sides of the triangle. The three basic trigonometric ratios are sine, cosine, and tangent, which are commonly abbreviated as sin, cos, and tan, respectively.
The expression (sin 3x)(cos x) − (cos 3x)(sin x) is an example of a trigonometric identity known as the "sin of a difference" formula:
sin(A - B) = sin A cos B - cos A sin B
By comparing this formula to the given expression, we can see that:
A = 3x
B = x
Substituting these values into the formula, we get:
sin(3x - x) = sin 2x = (sin 3x)(cos x) − (cos 3x)(sin x)
Hence, the given expression simplifies to sin 2x.