Sum and Difference Identities

Sum and Difference Identities

The sum and difference identities

(29)

(19)

are demonstrated by the following.

Fig. 1

Considering only the circumstances of the unit circle, as circles of other radii differ only by factors of r, we investigate a chord of length L, shown in Fig. 1, with an end point at (1,0). The other end point is given by (cos a, sin a), and its length is given by

(1)
(2)
(3)
(4)
(5)

Placing an arbitrarily positioned chord also of length L and labeling such that the angle u is greater than the angle v,

Fig. 2 this time L is given by

(6)
(7)
(8)
(9)
(10)

Equating 5 and 10 by their left sides,

(11)

where straight forward algebra produces

(12)

Now, recognizing, with some help from Fig. 2, that the angle a is the difference between angles u and v,

(13)

we make the substitution, producing

(14)

This provides cosine's difference identity. For the sum identity, we apply this, replacing v with its negative.

(15)

Noting the basic trigonometric identities

(16)
(17)

we substitute to find

(18)

This provides cosine's sum identity. Finally combining both from eqns. 14 and 18, we get cosine's sum and difference identity.

(19)

For sine's sum and difference identity, we start with this identity for cosine, rewritten as

(20)

and the cofunction identities.

(21)
(22)

Applying these identities to eqn. 20 to replace all a terms with pi/2 - a

(23)

and renaming with

(24)

we get

(25)

Next renaming with

(26)

we have

(27)

again noting the basic identities

(16)
(17)

we get

(28)
(29)

That's sine's sum and difference identity.