 So, while there are many trigonometric identities, the only ones worth memorizing besides the Pythagorean identities are the sum and difference identities. And these are important for the following reasons. We know the exact values of sine and cosine for a few angles. So we know the sine and cosine of, well, we have to say it, zero degrees, thirty degrees, forty-five degrees, sixty degrees, ninety degrees. But you should actually know these as the sine and cosine of zero, pi over six, pi over four, pi over three, and pi over two radians. We can use these as reference angles to find other sine and cosine values. But what about others? For more angles, we have what are known as the angle, sum, and difference identities. For any angles A and B, the sine of A plus or minus B is the sine of A times the cosine of B plus or minus the cosine of A times the sine of B. Meanwhile, the cosine of A plus or minus B is the cosine of A, cosine of B, minus or plus the sine of A, sine of B. So remember that the plus or minus or the minus or plus is a compact way of writing two equations. So if we expand that out a little bit, the sine of A plus or minus B, well, that refers to the sine of A plus B and the sine of A minus B. And the equation, sine A cosine B plus or minus cosine A sine B, well, that's sine of A cosine B plus cosine A sine B and sine of A cosine B minus cosine A sine B. Likewise, cosine A plus or minus B, well, that's cosine A plus B, cosine A minus B. And the important thing to notice here is that this is minus or plus. The minus equation goes first, followed by the plus equation. These identities are important enough that they are one of the few things that you should definitely memorize. And here's a couple of important tricks to help you. First, the sine identity, if it's A plus B, it's plus. If it's A minus B, it's minus. Cosine switches. A plus B gives you the minus. A minus B gives you the plus. Sine mixes things up. So sine of A plus B will have sine A, cosine B, cosine A, sine B. Cosine, on the other hand, keeps everything separated. Cosine A, cosine B, sine A, sine B. For example, let's find an exact value for the cosine of 15 degrees and let's use, for variety, two different methods. So let's pull in our angle, sum, and difference formulas. And we can find the cosine of a sum or difference provided we know the cosines and the sines of the things that we're adding or subtracting. So we do know some of these sine or cosine values. So our goal is we'd like to find 15 degrees as the sum or difference of things we know the sine or cosine of. So after a little reflection, one possibility is we might find the cosine of 15 degrees to be the cosine of 60 degrees minus 45 degrees. So that's the cosine of a difference. So that'll be the cosine of the two plus the sine of the two. So that's cosine 60, cosine 45, plus sine 60, sine 45. And we know the cosine and sine of 60 and 45 degrees, so we'll substitute those values in to find our cosine of 15. A second way we might get the cosine of 15 degrees is as the cosine of 45 degrees minus 30 degrees. So the cosine of that difference will be then substituting our values gives us, well, we really should be using radians. So let's try to find the sine of 7 pi 12ths. So unfortunately, this does require us to deal with fractions. Fortunately, we know how to handle fractions. So we might begin by noticing that pi over 6 is 2 pi 12ths. Pi over 4 is 3 pi 12ths, and pi over 3 is 4 pi 12ths. Now we want to get 7 pi 12ths as the sum or difference of these values whose sine and cosine we know. And so one way we can get that is to write 7 pi 12ths as, which is pi over 4 plus pi over 3. So the sine of 7 pi over 12 is the same as sine of pi over 4 plus pi over 3. And the sine of a sum is the sine times the cosine plus the cosine times the sine. So we can rewrite our expression, substitute in the sine and cosine values, and get our value.