Thread: Proof of Compound Angles?

Could someone please give me the proofs, or point me to a website, of the compound angles formulae?
That is:
cos(A+B) = cosA.cosB - sinA.sinB

And the other ones, cos(A-B), sin(A+B), sin(A-B), tan(A+B), tan(A-B).

Are they just geometrical proofs, or are there algebraic ones? Thank-you.

Originally Posted by qspeechc

Could someone please give me the proofs, or point me to a website, of the compound angles formulae?
That is:
cos(A+B) = cosA.cosB - sinA.sinB

And the other ones, cos(A-B), sin(A+B), sin(A-B), tan(A+B), tan(A-B).

Are they just geometrical proofs, or are there algebraic ones? Thank-you.

The only proofs I have seen are geometric.

-Dan

The formal mathematical proof uses power series and the Cauchy product formula. But I doubt that is how you want it done.

Thank-you for the link topsquark, and TPH, I would like to see the power series proof, and you are correct in assuming that I do not know about Cauchy product formulae.

Originally Posted by qspeechc

Thank-you for the link topsquark, and TPH, I would like to see the power series proof, and you are correct in assuming that I do not know about Cauchy product formulae.

The Power series proof is to know that,


You can to show,

Now substitute those formulas and use the Cauchy product:
.