Reduction Identities

The reduction identities

(7)

(8)

are demonstrated by the following.

Fig. 1

Consider two vectors from the origin to points on a circle at angles of plus and minus theta, as shown in Fig. 1. From the definitions of sine and cosine

(1)
(2)

then

(3)
(4)

and

(5)
(6)

Combining eqns. 3 and 5 and combining eqns. 4 and 6 produces the reduction identities.

(7)
(8)