Because the siding laps bottom over top, it is angled to the vertical plane, which means that a straight 45* butt cut is not going to work, you must calculate the compound angle to accommodate the vertical angle and the corner angle.

I can't tell you how to calculate the proper angle, but I can tell you to use the pieces you're removing as a template to set your saw and a jig for cutting the new pieces. The corner angle will be easy, simply set your saw over the end of the board and tilt it to match the angle. The vertical angle won't be too bad to jig up either, but it will take some time to get it right and you'll need a right and left handed jig to cut both sides of the corner. What I would do is place a board tight along the top and bottom of the template piece, set your saw blade and leave the saw in full contact with the end cut. Now place a third board along the edge of the saw's shoe and screw it to the boards along the top and bottom of the template piece.

Now all you have to do is set the jig over the end of the piece of siding and make your cuts accordingly. Do a few test pieces and adjust your blade angle and jig angles accordingly.