Question

is cos(x+y)+cos(x-y)=2cosx(cosy) an identity?

Answer 1

Find out...
if you can turn the left-side into the right side using a series of conversions, then it's an identity...

In English, it means that regardless of what permissible values of x and y we put, the equality holds...

(compare for instance, with, 3x + 9 = 12, which is not true for all values of x, rather, only for x = 1 )

Answer 2

Okay... we have these known identities that will help...
\[\Large \cos(\color{red}x+\color{blue}y) = \cos(\color{red}x)\cos(\color{blue}y) - \sin(\color{red}x)\sin(\color{blue}y)\]\[\Large \cos(\color{red}x-\color{blue}y) = \cos(\color{red}x)\cos(\color{blue}y) + \sin(\color{red}x)\sin(\color{blue}y)\]

Answer 3

So for the left-side, we have
\[\Large \cos(\color{red}x+\color{blue}y) +\Large \cos(\color{red}x-\color{blue}y)\]

Using the identities shown, these two can be expanded.