Using the Sine and Cosine Addition Formulas to Prove Identities
:Applying the cosine addition and sine addition formulas proves the cofunction, add pi, and supplementary angle identities. Using the formulas, we see that sin(pi/2-x) = cos(x), cos(pi/2-x) = sin(x); that sin(x + pi) = -sin(x), cos(x + pi) = -cos(x); and that sin(pi-x) = sin(x), cos( -x) = -cos(x). The formulas also give the tangent of a difference formula, for tan(alpha-beta).
Let's review the cosine addition formulas and the sine addition formulas. By cosine addition formulas I mean both the cosine of a sum and a cosine of the difference and remember that for both of these it's cosine cosine, sine sine. Cosine alpha cosine beta, sine alpha sine beta. With the cosine of a sum, you change the sine with cosine you always change sine think c stands for change, again cosine cosine, sine sine, the minus becomes plus. Now for the sine addition formulas, it's sine cosine cosine sine, so sine alpha cosine beta, it's beta, cosine alpha sine beta and the sine stays the same think s for the same plus sine alpha cosine beta cosine alpha sine beta minus stays minus. So these are the cosine addition formulas and the sine addition formulas.
And what's really we neat about these is an addition of being able to calculate exact values of certain special angles. We can use these to prove identities and I want to start with the cofunction identities for an example. Now you probably already know these identities but it's great to help you remember them to be able to prove them using the sine addition or subtraction formulas, so the sine of pi over 2 minus theta. This is this is going to give us a cofunction identity, let's use the sine of a difference formula. It's sine cosine, sine pi over 2 cosine theta cosine sine, cosine pi over 2 sine theta because it's minus and sine sine says the same, we have a minus. Now sine pi over 2 is 1 times cosine theta, cosine pi over 2 is 0, 0 times sine theta so this just becomes cosine theta and that's the formula we'll remember. Sine of pi over 2 minus theta equals cosine theta.
Likewise here's cosine pi over 2 minus theta, the cosine of a difference formula is what we need here. The cosine of a difference formula goes cosine cosine sine sine. Cosine pi over 2 cosine theta sine pi over 2 sine theta and cosine the sines change c for change so minus becomes plus. Remember cosine of pi over 2 0 sine of pi over 2 is 1 and so we end up with sine theta. Cosine of pi over 2 minus theta equals sine theta, so the cosine and sine addition formulas are really useful for proving identities.