Expert: Paul Klarreich - 2/1/2007

Question
Hi
I was wondering how to prove any of the four sum to product identities. I'm not sure where to begin.


sinA + sinB=2sin(A+B/2)cos(A-B/2)

sinA - sinB=2cos(A+B/2)sin(A-B/2)

cosA + cosB=2cos(A+B/2)cos(A-B/2)

cosA - cosB=-2sin(A+B/2)sin(A-B/2)  

Answer
Questioner:   Melissa
Category:  Advanced Math
 
Subject:  Trigonometry
Question:  Hi
I was wondering how to prove any of the four sum to product identities. I'm not sure where to begin.

sinA + sinB=2sin(A+B/2)cos(A-B/2)

sinA - sinB=2cos(A+B/2)sin(A-B/2)

cosA + cosB=2cos(A+B/2)cos(A-B/2)

cosA - cosB=-2sin(A+B/2)sin(A-B/2)
..............................
Hi, Melissa,

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

Start by letting:
   A + B
x = -----
     2
   A - B
y = -----
     2
Then solve these equations for A and B:

2x = A + B
2y = A - B

Add:      2A = 2x + 2y,  so  A = x + y
Subtract: 2B = 2x - 2y,  so  B = x - y.

Now try the first one.

sin A + sin B = sin(x + y) + sin(x - y)
Now apply the reduction formulas for those:

sin(x + y) = sin x cos y + cos x sin y
sin(x - y) = sin x cos y - cos x sin y

sin A + sin B = sin(x + y) + sin(x - y) =

sin x cos y + cos x sin y + sin x cos y - cos x sin y
= 2 sin x cos y
        A + B       A - B
= 2 sin (-----) cos (-----)
          2           2

After this, I think you can handle the others.  They might have tricky minus signs, but you'll be careful, I'm sure.