So you have the equations and , and you want to eliminate t between them.
Write the first equation as and think of and as sides in a right-angled triangle. Pythagoras' theorem tells you that and , where . Substitute those values for cos(t) and sin(t) into the second equation, do a bit of cancelletion, use the fact that , and it simplifies to . Now square both sides to get , and a bit more simplification reduces that to .