### Video Transcript

Find the Cartesian equation of the
curve defined by the parametric equations π₯ equals two plus cos π‘ and π¦ equals
four cos of two π‘.

Remember, a Cartesian equation is
one which contains only the variables π₯ and π¦. So weβre going to need to find a
way to eliminate our third variable π‘ from our parametric equations. And at first glance, it doesnβt
seem to be a nice way to do so. But we can begin by recalling some
trigonometric identities. We have cos of two π‘ in our second
parametric equation. And we know that cos of two π‘ is
equal to two times cos squared π‘ minus one. This means we can rewrite our
equation for π¦ as four times two cos squared π‘ minus one.

Next, weβll look to rearrange our
equation for π₯ to make cos of π‘ the subject. Once weβve done that, weβll be able
to find an expression for cos squared π‘ in terms of π₯. We can subtract two from both
sides. And we see that π₯ minus two equals
cos of π‘. Then, by squaring both sides of
this equation, we find that cos squared π‘ is equal to π₯ minus two all squared. And so, weβre now able to replace
cos squared π‘ with π₯ minus two squared. That gives us π¦ equals four times
two times π₯ minus two all squared minus one.

We distribute this first pair of
parentheses. And we find that π₯ minus two all
squared is equal to π₯ squared minus four π₯ plus four. We distribute again by multiplying
each of these terms by two and then simplifying: eight minus one is seven. Well, finally, we distribute one
more time by multiplying each term of two π₯ squared minus eight π₯ plus seven by
four. And we find the Cartesian equation
of the curve defined by parametric equations π₯ equals two plus cos π‘ and π¦ equals
four cos of two π‘ is π¦ equals eight π₯ squared minus 32π₯ plus 28.