 If we let A equal B in our angle sum formulas for sine and cosine, we get... Consequently... Which are basically useless since there are special cases of the angle sum formulas. Remember, understand concepts. Don't memorize formulas. More usefully, what if we want to do something like find the sine of 15 degrees? Remember, a problem exists even if we don't know how to solve it. So we note that the sine of 15 degrees shows up in the expansions of both cosine of theta plus or minus 15 and sine of theta plus or minus 15. These also require knowing the value of cosine and sine as well, so let's make both angles 15 degrees. Let's use the formula for the cosine of a sum. So the cosine of 15 plus 15 will be... Which we can simplify. Now notice that our right-hand side includes the square of sine and cosine, and remember the fundamental trigonometric identity, which gives us a relationship between the two, and so we can replace one with the other. So let's solve for cosine squared, replace, and simplify, and figure out the value of cosine 30. And now the only thing we don't know in this equation is the sine of 15 degrees. So solving for the sine of 15 degrees. And because we're taking a square root, we will have to choose which square root we want. And since 15 degrees is a first quadrant angle, the sine is positive and we'll choose the positive square root. Every formula in mathematics comes from automating a process. So we started with the cosine of 2 theta. We used our Pythagorean identity to solve for cosine squared and replace. Then we solved our equation for the sine of theta, and we had to decide which square root to use. One of the secrets for success in life, and in mathematics, is ask what happens if. So we eliminated the cosine squared, if instead we substituted for sine squared, again our Pythagorean identity would allow us to rewrite our cosine 2 theta equation as then solving for cosine theta would give us, and again we still have to decide, positive or negative square root. And putting our results together, give us the half angle formulas. But remember, I'll never mind you're just going to memorize the formulas anyway. But you should understand that they do come from the angle sum identities plus a little bit of algebra. So let's find the sine of 15 degrees using our formula. Well, that's actually what we did. And all our formula did is it allowed us to skip the algebra, which is in fact the whole point of having a formula. How about the cosine of 105 degrees? And we know we've already found this twice using the cosine of a sum as 60 plus 45, and the cosine of a difference using 135 minus 30, and we got an answer. Let's use the half angle formula and see if we get the same result. So we'll drop it into our formula. Now the reason this works is that 2 times 105 is 210 degrees, and we know the cosine of 210. And since 105 degrees is an angle in the second quadrant, cosine of 105 is negative, and so we conclude. Uh oh, that's a different result. But if we compute the values we do find that the two wildly different expressions do represent the same number, so we do get the same answer.