Question 1134076
<pre>
Use the two identities:

{{{cos(x+y) = cos(x)cos(y)-sin(x)sin(y)}}}

{{{cos(x-y) = cos(x)cos(y)+sin(x)sin(y)}}}

Multiply left sides and right sides:

{{{cos(x+y)cos(x-y) = (cos(x)cos(y)^""-sin(x)sin(y))(cos(x)cos(y)^""+sin(x)sin(y))=""}}}

Multiply out the right side 

{{{cos^2(x)cos^2(y)-sin^2(x)sin^2(y)=""}}}

{{{cos^2(x)(1-sin^2(y)^"")-(1-cos^2(x)^"")sin^2(y)=""}}}

{{{cos^2(x)-cos^2(x)sin^2(y)-sin^2(y)+cos^2(x)sin^2(y)=""}}}

{{{cos^2(x)-sin^2(y)}}} <-- possible final answer

That may be far enough for your teacher, but you can keep 
going if you desire.  Use these two identities:

{{{cos(2x)=2cos^2(x)-1}}} and {{{cos(2y)=1-2sin^2(y)}}}

Solve them for the squared terms.

{{{2cos^2(x)=1+cos(2x)}}} and {{{2sin(2y)=1-cos^2(y)}}}

{{{cos^2(x)=1/2+expr(1/2)cos(2x)}}} and {{{sin(2y)=1/2-expr(1/2)cos^2(y)}}}

Substitute in

{{{cos^2(x)-sin^2(y)=""}}}

{{{(1/2+expr(1/2)cos(2x))-(1/2-expr(1/2)cos(2y))=""}}}

{{{1/2+expr(1/2)cos(2x)-1/2+expr(1/2)cos(2y)=""}}}

{{{expr(1/2)cos(2x)+expr(1/2)cos(2y)=""}}}

{{{expr(1/2)(cos(2x)^""+cos(2y))}}}

Edwin</pre>