Product to sum formulas for sin and cos - PROOF

Derivation/explanation of the product-to-sum identities for sine and cosine. We've got some trig today!

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[ethanmullen]

Proving trig identities from other trig identities feels so wrong...but so right

[amrshamsel-deen]

Way to go
That Helped alot
in Fourier series

[doggydoggywho]

This was so great. To the point and clear.

[fabiojr]

Thanks! How about a formula for tan(x)tan(y)?

[patricksalhany]

Nice.
I like to go to the complex world and use Euler's formulas for cos and sin.

[josepineda]

Thanks for the help, pretty easy explanation

[sayote]

I just wanted to ask how can sin(a)cos(b )= sin(b)cos(a) and cos(a) doesn’t equal cos(b) nor does sin(a) equals sin(b)