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**

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