The keys are these two formulas:

      = ,        (1)   and

      = .        (2)

(see the lesson Powers of trigonometric functions in this site).  When applying them, you will get

cos^3(A)*cos(3A) =  ,   (3)  and

sin^3(A)*sin(3A) = .   (4)


So, adding and expanding (3) and (4), you will get

   = 

=  = 

= [] + [] = 

    For the first  bracket  [ . . ]  apply the formula cos(2x) = . . .   
    For the second bracket  [ . . ]  apply the formula cos(x-y) = . . .   You will get

=  =  = 

    Now apply again the formula (1). You will get

= .