Sum them together as C+i*S and write in exponential form. You'll get a GP. Find the sum. Separate the real and imaginary parts.
I've not done the whole thing so I make no guarantees.
how do i calculate the sum of each of these series?
sin(A) + sin(A+B) + sin(A+2B) + sin(A+3B) .... + sin(A+(n-1)B)
cos(A) + cos(A+B) + cos(A+2B) + cos(A+3B) .... + cos(A+(n-1)B)
please show step by step how to do it, not just the answer.