Trigonometry

sin (a + b) = sin acos b + cos asin b.

(A.4)

sin (2a) = 2sin acos a.

(A.5)

cos (a + b) = cos acos b − sin asin b.

(A.6)

cos (2a) = {cos}^{2} a − {sin}^{2} a.

(A.7)

From Eq. (A.7) and ${cos}^{2} a + {sin}^{2} a = 1$ :

{cos}^{2} a = \frac{1}{2} (1 + cos (2a))

(A.8)

and

{sin}^{2} a = \frac{1}{2} (1 − cos (2a)).

(A.9)

The following are some of the so-called product to sum identities:

cos (a)cos (b) = \frac{1}{2} [cos (a − b) + cos (a + b)].

(A.10)

sin (a)sin (b) = \frac{1}{2} [cos (a − b) − cos (a + b)].

(A.11)

sin (a)cos (b) = \frac{1}{2} [sin (a + b) + sin (a − b)],

(A.12)

where $a$ is the argument of the sine in Eq. (A.12).