verify the following identity
cos(x+y) + cos(x-y)=2cosxcosy
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
What happens when you add the two together?
To verify the given identity, we'll start with the left side of the equation and try to manipulate it to match the right side.
Left side of the equation:
cos(x+y) + cos(x-y)
Using the trigonometric identity for the sum of angles, we can rewrite the first term:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Using the trigonometric identity for the difference of angles, we can rewrite the second term:
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
Substituting these values back into the left side of the equation, we have:
(cos(x)cos(y) - sin(x)sin(y)) + (cos(x)cos(y) + sin(x)sin(y))
Rearranging the terms, we get:
cos(x)cos(y) + cos(x)cos(y) - sin(x)sin(y) + sin(x)sin(y)
Combining like terms, we have:
2cos(x)cos(y)
Now, we can see that the left side of the equation is equal to the right side of the equation:
cos(x+y) + cos(x-y) = 2cos(x)cos(y)
Therefore, the identity is verified.