Square board

You are in front of a 8 by 8 square board (so it has 64 squares). You also have 32 rectangle pieces whose size are 1 by 2 (they fit on exactly two squares of the board). Obviously you can cover the board with the pieces. Now, it give you another board, also 8 by 8, but with the top left and the bottom right squares removed. I also remove one of your 32 pieces.

Can you still cover the whole board and why?

Let's say that the board is painted like a chess board (with black and white squares). Then, no matter how you cover the board, each piece will be one black square and one white square. The two removed squares being of the same color, the covering that was trivial before is not impossible.