gistfile1.txt

For given values of x and y, define the following function of a real parameter t:

    f(t) = cos(tx) cos(x + y - tx) - sin(tx) sin(x + y - tx)

Note that f(0) = cos(x + y) and f(1) = cos(x) cos(y) - sin(x) sin(y), so this is a
smooth deformation between the two sides of the cosine addition formula. The addition
formula will follow if we can prove df/dt = 0. Let us compute:

    d/dt cos(tx) cos(x + y - tx) = -sin(tx) cos(x + y - tx) + cos(tx) sin(x + y - tx))
    d/dt sin(tx) sin(x + y - tx) =  cos(tx) sin(x + y - tx) - sin(tx) cos(x + y - tx)
    
These two expressions are equal and hence df/dt = 0.