Proof of the Sum-Difference Formulas - Part 1

In this video, I demonstrate how to prove the following sum-difference formulas, or trigonometric identities:

cos(a - b) = cos(a)*cos(b) + sin(a)*sin(b)
cos(a + b) = cos(a)*cos(b) - sin(a)*sin(b)
sin(a + b) = sin(a)*cos(b) + sin(b)*cos(a)
sin(a - b) = sin(a)*cos(b) - sin(b)*cos(a)

From the results above, I also derive the following identities:

cos(2a) = cos^2(a) - sin^2(a)
sin(2a) = 2*sin(a)*cos(a)

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cos(x - y) = cos(x)*cos(y) + sin(x)*sin(y)
cos(x + y) = cos(x)*cos(y) - sin(x)*sin(y)
sin(x + y) = sin(x)*cos(y) + sin(x)*cos(y)
sin(x - y) = sin(x)*cos(y) - sin(x)*cos(y)
cos(2x) = cos^2(x) - sin^2(x)
sin(2x) = 2*sin(x)*cos(x)