 In this video we provide the solution to question number nine for the practice final exam for math 1060 in which case Using the substitution x equals three sine of theta We need to rewrite the expression cosine of two theta over four algebraically in terms of just the variable x right there So starting off with this substitution We actually want to think of a triangle diagram associated to theta right here, and this would be a right triangle of course Where the angle theta comes into play here well if x equals three sine theta that actually means that sine theta is Equal to x over three. This is the opposite over hypotenuse relationship Filling in the third side by the Pythagorean equation The adjacent side would be three squared, which is nine minus x squared like so and so using this triangle We can write any trigonometric expression. What about cosine of two theta over four? Well, we have one fourth cosine of two theta Like so well, I don't have a triangle diagram for two theta have a triangle diagram for theta So maybe we can use some type of trigonometric identity to help us out in a situation like this And that's exactly the case cosine of two theta Well the double angle identity for cosine and we could use cosine squared minus sine squared Or there's a couple of variations. Well, since we know sine I'm actually going to prefer to use that one right now one fourth times one minus two sine theta sine theta like so Sine squared theta excuse me and so honestly, I could have gotten away with this how without having to draw the diagram whatsoever because Of the formula just depends on sine right here So we end up with one fourth times one minus two times well x over three Square that of course and so we end up with one minus Don't forget the one fourth there one minus We're gonna get two x squared over nine and then times everything by four We end up with one fourth minus two x squared over four times nine There's a common factor of two right there And so we end up with one fourth minus x squared over 18 And so we see that the correct answer would then be choice C One fourth minus x squared over 18