I want to derive the cosine of the sum formula. We already derived the cosine of the difference formula all we need to derive the cosine of the sum. Our two identities, these are the opposite angle identities for cosine and the sine. Remember that the cosine of the opposite of theta equals the cosine of theta. The cosine is an even function, opposite inputs give the same output.
And remember that opposite inputs give opposite outputs for sine, we'll be using this in the next proofs. So cosine of a sum what I'm looking for is the cosine of alpha plus beta. Now I already proved, I already found a formula for the cosine of the difference, so all I really need to do is express this as the difference; cosine of alpha minus negative beta, and you recall that the formula was cosine, cosine, sine, sine cosine alpha, cosine negative beta, sine alpha, sine negative beta and you remember that the minus sign becomes a plus sign in the cosine formula.
So here is what I use the odd even identities, the opposite angle identities. Cosine of negative b is the same as cosine of b, beta sorry, beta is the Greek letter b and then sine of alpha sine of negative beta becomes minus the sign of beta, so I get a minus sine beta and I'm done. This is the cosine of alpha plus beta, this is the cosine of a sum, it's cosine cosine, sine, sine just like the cosine of difference, but the plus becomes a minus, just remember 'c' change sign. Let's use this in an example really quickly.
It says find the exact values of cosine of 75 degrees. What I need to do is find two special angles that add up to 75 degrees like 30 degrees and 45 degrees. So 30 degrees plus 45 degrees. Now and remember cosine, cosine, sine, sine. Cosine 30 degrees cosine 45 degrees, sine 30 degrees, sine 45 degrees and change the sign, the plus becomes a minus and the rest is just make sure you've got these memorized, these values. Cosine of 30 degrees root 3 over 2, cosine of 45, root 2 over 2 minus sine of 30 degrees one-half and sine of 45, 2 over 2, so you get root 6 over 4, minus root 2 over 4, that's root 6 minus root 2 over 4 and that's your answer. That's the cosine of 75 degrees.