 Consider the following diagram. We have the unit circle illustrated right here. We have a line or I should say a ray emanating from the origin that then terminates and forms an angle, which we're going to call theta right here. There are two right triangles in play right here. There's the triangle OAP, which has its sides relative to theta. The opposite side is sine, the adjacent side is cosine, and the hypotenuse is one. We also have the triangle OBR, which again, relative to the angle theta, we have the side OB, which has length one. We have the opposite side BR, which is unknown at the moment. And then we have the hypotenuse OR, which is also unknown at the moment. We want to argue using similar triangles that the hypotenuse of that larger triangle OR is actually equal to secant theta. So again, we're going to do this with a similar triangle type argument here. So we have the angle theta. It's a right triangle like so. This is O, this is A, and this is P. So because P is a point on the unit circle, we know that the side OA is cosine of theta. We know side PA is sine of theta. And again, because it's a unit circle, the hypotenuse will be one because it's a radius of the unit circle. The other triangle, we'll draw this one in blue, this is the triangle OBR. It has angle theta. It's also a right triangle. The side OB, because it's the radius of the unit circle, it's length will be one. RB is not told to us, it's not labeled. It is tangent, but we actually don't need that. We won't use that. So we want to compute OR and claim that that is in fact secant. And again, we're going to do this by similar triangles. So first note that the triangle OAP is similar to the triangle OBR. How do we know that? Well, they both have angle theta. They both have a right angle. And so the remaining angle, since all angles sums that up to 180 degrees for these triangles would have to be the same. So these triangles have the same angle measurements. So they are similar to each other. And so since the triangles are similar, you do need to mention that these triangles are similar. Because they're similar triangles, we can set up proportions. So we want to set up a proportion so that we can find OR. So we're going to take OR, which is what we don't know. We want to show it's equal to secant. We have to then take a second side of the triangle. The only other side we know is OB. So we're going to take the segment OB. We did have to set this equal to the corresponding positions. OR is the hypotenuse of this triangle. So it'll compare to one. So then say that this compares to OP. And then to OP right there. And then the side OB corresponds to OA. So we then put those there as well. Now, OP we know is one. And OA we also know to be cosine of theta. So since OB is one, this then shows that OR is equal to one over cosine theta, which that's exactly the same thing as secant.