We have to prove that 

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

From the figure,

The angle of the upper triangle i.e. opposite side of the length C is x-y.

Now by cosine law,

      C² = 1² + 1² - 2*1*1*cos(x-y)

=> C² = 1 +1 - 2*cos(x-y)

=> C² = 2 - 2*cos(x-y)

Agian the side of length C joins the points (cosy, siny) and (cosx, sinx), So from Pythagorus theorem

      C² = (cosy - cosx)² + (siny - sinx)²

=> C² = cos² y + cos² x - 2*cosx*cosy + sin² x + sin² y - 2*sinxsiny

=> C² = (cos² y + sin² y) + (cos² x + sin² x) - 2(cosx*cosy - sinxsiny)

=> 2 - 2*cos(x-y) = 1 + 1 - 2(cosx*cosy - sinxsiny)                              (since cos² θ + sin² θ = 1, C² = 2 - 2*cos(x-y) )

=> 2 - 2*cos(x-y) = 2 - 2(cosx*cosy - sinxsiny)

=> - 2*cos(x-y) = - 2(cosx*cosy - sinxsiny)

=>  cos(x-y) = cosx*cosy - sinxsiny