Write the product as a sum or difference containing only sin

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

Write the product as a sum or difference containing only sines or cosines. Cos (theta/2) sin (9 theta/2) Cos (theta/2) sin (9 theta/2) =

cos(theta/2)sin(9theta/2)

cos(theta/2)cos(pi/2 - 9theta/2)

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

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

On comparing we get

A+B/2=theta/2 and A-B/2= pi/2 - 9theta/2

Solving them we get

A= (pi/2) - 4theta and B= 5theta - (pi/2)

So we get (cos (pi/2 - 4theta)+ cos (5theta-(pi/2)))/2

(sin 4 theta + sin 5theta)/2