Prove: cos(x+y)cosy + sin(x + y)siny = cosx

cos(x+y)*cosy + sin(x+y)*siny =
cosx

First we will use trigonometric
identities.

We know
that:

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

sin(x+y) = sinx*cosy +
sinx*cosy

Now we will susbtitute into the
equation.

==> [cosx*cosy - sinx8siny) cosy +
(sinxcosy + cosx*siny)siny

We will expand the
brackets.

==> cosx*cos^2 y - sinx*siny*cosy +
sinx*siny*cosy + cosx*sin^2 y

We will reduce similar
terms

==> cosx*cos^2 y + cosx*sin^2
y 

Now we will factor cosx from both
terms.

==> cosx*(cos^2 y + sin^2
y)

But sin^2 y + cos^2 y =
1

==> cosx*1 = cosx.................
q.e.d