Rewrite sin(6x) sin(x) as a sum or difference.

cos(6x - x)  = cos6xcosx + sin6xsinx   (1)

cos(6x + x) = cos6xcosx - sin6xsinx     (2)

Subtract   (2) from (1)   and we have  :

cos(6x - x) - cos (6x + x) = sin6xsinx + sin6xsinx   .....  simplify

cos(5x) - cos(7x)  = 2sin6xsinx

2sin6xsinx = cos(5x) - cos(7x)

sin6xsinx  = (1/2)[cos(5x) -  cos(7x) ]

 [Thanks to Melody for spotting an earlier couple of typos  !!! ]

I had an easier answer

cos(6x - x)  = cos6xcosx + sin6xsinx 

so

sin6xsinx = cos5x  - cos6xcosx

That is a difference