`cos(x + y) cos(x - y) = cos^2(x) - sin^2(y)` Prove the identity.

`cos(x+y)cos(x-y)=cos^2(x)-sin^2(y)`

We will use the following product formulas to prove the identity,

`2cosAcosB=cos(A+B)+cos(A-B)`

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

`=(cos(x+y+x-y)+cos(x+y-(x-y)))/2`

`=(cos(2x)+cos(2y))/2`

Now we will use`cos(2theta)=2cos^2(theta)-1, cos(2theta)=1-2sin^2(theta)`

`=(2cos^2(x)-1+1-2sin^2(y))/2` 

`=(2(cos^2(x)-sin^2(y)))/2` 

`=cos^2(x)-sin^2(y)`

LHS=RHS , Hence proved.