Right

You have found the value for cos(2y), but that wasn't asked for.

Good.

Looks like you're trying to find the value of cos(x/2), which wasn't asked for. If so, the 2 belongs inside the square root sign, so it should be 3/sqrt(10)

You have siny = -12/13, but are you sure it isn't actually +12/13? Your post is a bit garbled regarding what quadrant y is in, so I can't really tell whether this is correct or not.

No, sin(x-y) = sinx cosy - cos x siny