10 Points! Help proving Trig Identities

1. tan(x+y)tan(x-y) = (sin^2x - sin^2y) / (cos^2x - sin^2y)

Also just a side question, does Tan2x = sin2x / cos2x? and 2tanx = 2sinx / 2cosx ?

tan(x+y)tan(x-y)

=

sin(x+y) . sin(x-y)
----------- . -----------
cos(x+y). cos(x-y)

=

[sin(x)cos(y) + sin(y)cos(x)] . [sin(x)cos(y) - sin(y)cos(x)]
-------------------------------------- . -------------------------------------
[cos(x)cos(y) - sin(x)sin(y)] . [cos(x)cos(y) + sin(x)sin(y)]

=

sin²(x)cos²(y) + sin(x)sin(y)cos(x)cos(y) - sin(x)sin(y)cos(x)cos(y) - sin²(y)cos²(x)
---------------------------------------…
cos²(x)cos²(y) - sin(x)sin(y)cos(x)cos(y) + sin(x)sin(y)cos(x)cos(y) - sin²(x)sin²(y)

=

sin²(x)cos²(y) - sin²(y)cos²(x)
---------------------------------------…
cos²(x)cos²(y) - sin²(x)sin²(y)

=

sin²(x)[1 - sin²(y)] - sin²(y)[1 - sin²(x)]
---------------------------------------…
cos²(x)[1 - sin²(y)] - [1 - cos²(x)]sin²(y)

=

sin²(x) - sin²(x)sin²(y) - sin²(y) + sin²(x)sin²(y)
---------------------------------------…
cos²(x) - cos²(x)sin²(y) - sin²(y) + sin²(y)cos²(x)

=

sin²(x) - sin²(y)
----------------------
cos²(x) - sin²(y)

Ha! Phew... done

And on the side question, yes, it does. :)