Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Trigonometric identities. Tangent Theta equal sine Theta over cosine Theta, cotangent Theta equal 1 divided by tangent Theta or cosine Theta divided by sine Theta. Cosecant Theta is one over sine Theta and secant Theta is one over cosine Theta. If we thought about looking at a unit circle and we put on any point, xyx was the amount we went to the right and Y is the amount we went up, and the radius of a unit circle is always 1. So looking at this right triangle, we could see that cosine Theta is really X / 1 and sine Theta is really y / 1 because cosine is adjacent over hypotenuse and sine is opposite over hypotenuse. So in the circle we know that X ^2 + y ^2 equal 1 ^2. If we put in instead of X what it equaled, which was cosine Theta, we'd get a new equation saying cosine squared Theta plus sine squared Theta equal 1. This is one of three Pythagorean identities. The other two come directly from taking that equation cosine squared Theta plus sine squared Theta equal 1 and dividing each term through by cosine squared Theta. If I divide by cosine squared Theta, cosine squared over cosine squared is 1, sine squared over cosine is tangent squared. One over cosine squared is secant squared. So here is my second Pythagorean identity that says one plus tangent squared Theta equals secant squared Theta. I think about this one with the S and the T having to go together because they're close together in the alphabet. If we took the original equation and divided everything through by sine squared this time we'd get cotangent squared Theta plus one equal cosecant squared Theta. That's our third Pythagorean identity. I remember those by the C's go together. The cotangent cosecant go together. As long as you know the first one, you ought to easily be able to find the other two. We need to look at even and odd identities. If we look at our sine graph, Y equals sine X. When we have X, we know that the Y value is positive. If we had negative X, the Y value would be negative. So to get those two to be equivalent, we'd have the sign of negative X equaling the opposite of sine of X. So the negative Y value here would equal the opposite of the positive Y value, and the opposite of a positive is negative. That's the same for the cosecant graph. So cosecant and negative X equal negative cosecant X. These two are odd functions by odd function. If we picked it up. If we picked the graph up and pivoted it around the origin and put it down 180° later, it would look exactly the same as the original. For the cosine graph, if we went over positive X and we went negative X in the left direction, we'd see these Y values are actually the same. So cosine of negative X equal cosine X, secant of negative X equals secant of X. These are called even functions. And then even function is if I fold it along the Y axis, the left side will fold right on top of the right side tangent X. When we draw our graph here, we can see tangent X is going to give me out a positive Y value if I'm to the right and a negative Y value if I'm on the left. Remember I'm taking just any old generic X. So if I'd gone out even further I might have had an X value that was negative, but then the negative X value would be the opposite or positive. So tangent of negative X is going to equal the negative tangent X. And this is an odd function. Again, if we thought about pivoting around the origin by 180°, it would look just the same cotangent X. If we look at the positive direction off to the right and we just choose some value, we'd have an, we'd have AY that's positive. If we go to the left, we'd have AY that's negative. To get those two to be equivalent, we'd get cotangent negative X equaling negative cotangent X. This one is again an odd function. So when we do trigonometric identities, what we're going to do is we need to make one side equaling another side. So if I started with something like cosecant X minus cotangent Y equals sign X over 1 plus cosine X, this one was supposed to be an X. Well, we want to only manipulate one side, the left side or the right side. So if we looked at cosec an X, we know that that's one over sine. And if we look at cotangent, we know that that's cosine over sine. So now we have 1 minus cosine X over sine X. Well, that's not quite what we want. We want it to be sine X / 1 plus cosine X. So if we thought about doing a conjugate, the conjugate of 1 minus cosine X is one plus cosine X. If I multiply the top by one plus cosine X, I'm going to multiply the bottom, and one plus cosine X / 1 plus cosine X is really multiplying by 1. So we get one minus cosine squared X over sine X * 1 plus cosine X. Well, we've got that one plus cosine X in the bottom that we needed. We have a Pythagorean identity that says sine squared X plus cosine squared X equal 1. So sine squared X would equal 1 minus cosine squared X. So this one minus cosine squared XI can take out and put in sine squared X. Then I have that over sine X1 plus cosine X, the sine XS are going to cancel, leaving us just a plain sine X on top, and we get sine X / 1 plus cosine X. So that's one identity. We've just proved the left side equaling the right side. If we look at another one, we could have 1 minus cosine squared Theta times one plus cotangent squared Theta equals one. Well, by our Pythagorean identity, we know that sine squared Theta plus cosine squared Theta equal 1. So sine squared Theta is 1 minus cosine squared Theta. So this one minus cosine squared Theta I might replace a sine squared Theta. We also have an identity that says one plus cotangent squared Theta equal cosecant squared Theta. So instead of that one plus cotangent squared Theta, we might put in cosecant squared Theta and then we know that cosecant is really just one over sine. So if I have sine squared Theta times one over sine squared Theta, that equals 1. So 1 does equal 1 if we do another one, let's say secant Theta over cosecant Theta plus sine Theta over cosine Theta equal to tangent Theta. Well secant Theta is really one over cosine and cosecant is really one over sine. So if we simplify that first secant Theta over cosecant Theta, we really are going to get sine Theta over cosine Theta. And if I have sine Theta over cosine Theta plus sine Theta over cosine Theta, we get 2 sine thetas over cosine Theta. But sine Theta over cosine Theta is really just tangent. So 2 tangent Theta equal 2 tangent Theta left side equals the right side. What if we had the natural log of one plus cosine Theta plus the natural log of 1 minus cosine Theta equal to natural logs the absolute value of sine Theta? Well, if we have a natural log plus a natural log, we can rewrite it as a single natural log and we're going to multiply the terms. So one plus cosine times 1 minus cosine, which we realize is difference of squares. If we have a 2IN front, we can use the power rule to take that up and make it sine Theta that's now squared. So, oh, except we're only allowed to move one side, so let's leave that as two natural log sine Theta. If we continue to work on the left side, we get one minus cosine squared Theta and we have an identity that says that that's really just sine squared Theta. And then we can take the two by the power rule out in front. So 2 lane sine Theta equal 2 lane sine Theta. And that's true. Thank you and have a wonderful day.