Accordingly, sin(2a)=2sin(a)cos(a).

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

Accordingly, cos (2a) = (COSA) 2-(SINA) 2 = 2 (COSA) 2-1=1-2 (SINA) 2.

sin(a-b)= sin(a)cos(b)-cos(a)sin(b)

cos(a-b)= cos(a)cos(b)+sin(a)sin(b)

Then multiply and divide,

According to sin (a+b) = sin (a) cos (b)+cos (α) sin (b) and sin (a-b) = sin (a) cos (b)-cos (a) sin (b).

Add to get: (sin (a+b)+sin (a-b))/2 = sin (a) cos (b).

Where if (A+B)/2 is used to replace a and (A-B)/2 is used to replace b.

There is sin (a)+sin (b) = 2 sin ((a+b)/2) cos ((a-b)/2).