How do you form a Cartesian equation from two parametric equations?

If the two parametric equations have the form x = at + b and y = ct + d then the first step is to rearrange one to make the parameter 't' the subject. We then substitute this equation for 't' into the other parametric equation and rearrange to make y = f(x). In some cases, 't' may be raised to a power in either equation. It is usually quicker to start by rearranging the lowest order equation for 't' and substituting it into the higher order equation.

However, some questions may involve trigonometric functions e.g. x = sin^2(t) and y = cos(2t). We cannot simply rearrange these the same way. Instead we should list the associated trig identities for the functions involved. We see that cos(2t) = cos^2(t) - sin^2(t) = 1 - 2*sin^2(t) relates the equations for x and y alone. Substituting x and y in we find that y = 1 - 2x. The question will usually contain clues. For example, if cos(2t) is given then the double angle formula may be needed, hence the importance of listing all related identities by hand or mentally.