Exercise 6.9.15.

(a) Use the relationships sin(a+b) = sin

(a) cos(b)+cos

(a) sin(b)

and References 245 cos(a+b) = cos

(a) cos(b)−sin

(a) sin(b)
to show the following.
cos

(a) cos

(b) = 1 2 {cos(a+b)+cos(a−b)}.
cos

(a) sin

(b) = 1 2 {sin(a+b)−sin(a−b)}.
sin

(a) sin

(b) = 1 2 {cos(a−b)−cos(a+b)}.

(b) Recall that for complex numbers x with |x| < 1, nΣ
t=1 xt = x−xn+1 1−x .
Apply this fact to exp !
2πi jn "
to show that, for any j = 1, . . . ,n−1, nΣ
t=1 cos 
2π j n t 
= 0 = nΣ
t=1 sin 
2π j n t 
.

(c) Prove Eqs. (6.2.4), (6.2.5), and (6.2.6).