Write the product as a sum or difference containing only sin

Write the product as a sum or difference containing only sines or cosines.

Use a sum-to-product identity to rewrite the following as a product. Cos (pi/2 x + pi) + cos(pi x - pi/3) The sum cos(pi/2 x + pi) + cos (pi x - pi/3) is rewritten as a product. (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.)

Solution

cos(pi/2 + pi) + cos (pi x - pi/3)

cos A + cos B= 2cos(A+B)/2 cos (A-B)/2

So we get 2cos (pi x/2 + pi x + pi - pi/3)/2 cos (pi x/2 + pi - pi x + pi/3)/2

                 2 cos (3pi x/4    + pi/3 ) cos (2pi/3 - pi x/4)