Rewrite the product as a sum. sin(5x) sin(7x)

Answer 1

Since the problem has a product of two Sin values, we need to consider the Cos A minus Cos B identity in Trigonometry.

From that,

Cos A - Cos B = -2 * Sin[(A+B)/2] * Sin[(A-B)/2]

We can re-write the above as follows:

Sin[(A+B)/2] * Sin[(A-B)/2] = -1/2 * (Cos A - Cos B) ......... (1)

Further, to write this in terms of x, it must be true that

(A+B)/2 = 7x ---> A + B = 14x ...... (2)

and (A-B)/2 = 5x ---> A - B = 10x ...... (3)

Solving for A and B from the Equations (2) and (3) gives us

A = 12x and B = 2x

Putting those values for A and B back into Equation (1) gives us

Sin(7x)*Sin(5x) = -1/2 * (Cos(12x) - Cos(2x))

or

Sin(5x)*Sin(7x) = -1/2*Cos(12x) + 1/2*Cos(2x) ---- ANSWER