Question

How do you verify the identity `cos(x+y)+cos(x-y)=2cosxcosy`?

Answer 1

Remember your formula:

`cos(x + y) = (cosx * cosy) - (sinx*siny)`

Now, try this:

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

...so you can apply your formula again:

`= cosx * cos(-y) - sinx * sin(-y)`

Now here's the trick: remember that cosine is a symmetrical function about x = 0. This means that cos(-y) = cos(y) for all y.
Sine, however, is NOT symmetrical. sin(-y) = -sin(y) for all y.
(look at the graphs of these functions to verify this).

So you can rewrite `cos(x-y)` as:

`cosx * cosy - (sinx * (-siny))`
`= (cosx*cosy) + (sinx * siny)`

So therefore:
`cos(x + y) + cos(x - y) =`

`((cosx * cosy) - (sinx*siny)) + ( (cosx*cosy) + (sinx * siny))`

`= (cosx * cosy) + (cosx * cosy)`

`= 2(cosx * cosy)`

Answer 2

`"see explanation"`

Explanation:

`"using the "color(blue)"addition formulae for cosine"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(cos(A+-B)=cosAcosB∓sinAsinB)color(white)(2/2)|)))`

`"left side "`

`cos(x+y)+cos(x-y)`

`=cosxcosycancel(-sinxsiny)+cosxcosycancel(+sinxsiny)`

`=2cosxcosy=" right side "rArr" verified"`