Question

Prove the identity cos x + cos y = 2cos(x+y/2) cos(x-y/2).
a. Show that x+y/2+x-y/2=x
b. Find a similar expression using x+y/2 and x-y/2 that equals y.
C. Use parts (a) and (b) to prove the identity.

Answer 1

Answer:

Step-by-step explanation:

cos(c+d)=cos c  cos d-sin c sin d   ...(1)

cos(c-d)=cos  c  cos d+sin c sin d  ...(2)

add (1) and (2)

cos (c+d)+cos (c-d) =2 cos c cos d  ...(3)

put c+d=x

c-d=y

add

2c=x+y

c=(x+y)/2

subtract

2d=x-y

d=(x-y)/2

substitute in (3)

$cos~ x+cos~ y=2 cos (\frac{x+y}{2}) ~cos (\frac{x-y}{2})$