Why are the values for sine and cosine equal at 45°?

These are values that you should know by heart.  For other angles, you should be able to find the sine and cosine values using a calculator or a table.
                                           1
sin 45 deg = opp / hyp =
                                         sqrt(2)

cos 45 deg = adj / hyp = (same as above)
                                                                                sqrt(3)
Note that sin 30 deg = 1/2, and that cos 30 deg =
                                                                                     2