Starting with the cofunction identities, the sine addition formula is derived by applying the cosine difference formula. There are two main differences from the cosine formula: (1) the sine addition formula adds both terms, where the cosine addition formula subtracts and the subtraction formula adds; and (2) the sine formulas have sin-sin and cos-cos. Both formulas find values for angles.
I want to develop a formula for the sine of the sum of the of two angles. In order to do that I'm going to need to recall two identities, the cofunction identities. Remember the cosine of pi over 2 minus theta equals sine of theta and the sine of pi over 2 minus theta equals cosine theta. Remember that sine and cosine are cofunctions of one another and in order to transform one into the other all you have to do is replace theta by pi over 2 minus theta and pi over 2 minus theta is the compliment of theta so you're just replacing theta with its compliment. Anyway we're going to use those two identities on our proof and we're also going to use the cosine of a difference formula, so what's the sine of a sum? This is the kind of thing we're looking for the sine of alpha plus beta so first thing we do is use a cofunction identity to rewrite this, cosine and back over here again here my theta is going to be alpha plus beta I'm going to get cosine of pi over 2 minus alpha plus beta, so pi over 2 minus alpha plus beta and I'll just distribute the minus sign over the alpha and beta and I get cosine pi over 2 minus alpha minus beta and here you see that we actually have the the cosine of a difference. We've got pi over 2 minus alpha minus beta so we can use our formula for cosine of a difference and remember it's cosine cosine, sine sine, so it's cosine pi over 2 minus alpha cosine beta sine pi over 2 minus alpha sine beta and remember with cosine the minus becomes plus. Now we have two more situations where we have to use the cofunction identities. Cosine of pi over 2 minus alpha is sine alpha and sine of pi over 2 minus alpha is cosine alpha so sine alpha times cosine beta plus cosine alpha times sine beta and that's it, that's the sine of alpha plus beta. This is the sine of a sum identity.