The double angles sin(2theta) and cos(2theta) can be rewritten as sin(theta+theta) and cos(theta+theta). Applying the cosine and sine addition formulas, we find that sin(2theta)=2sin(theta)cos(theta). Also, cos(2theta)=cos2(theta) - sin2(theta), see other forms for the two derivations. These results reappear in integral calculus, when remembering them can be the difference between a right and wrong answer.
I want to use the sum formulas to introduce some new identities. Recall the sine of a sum alpha plus beta is sine of alpha cosine beta plus cosine alpha sine beta and the cosine of the sum alpha plus beta is cosine alpha cosine beta minus sine alpha sine beta.
Let's use these two identity identities to derive the double angle formulas first for sine. I'll call it sine of 2 theta now sine of 2 theta is the same as sine of theta plus a theta, so we just apply the sine of a sum identity here. We get sine cosine, cosine sine but that's just 2 sine theta cosine theta and that's our identity double angle identity for sine. Sine of 2 theta equals 2 sine theta cosine theta.
Let's do the same thing for cosine. Cosine 2 theta, the cosine of a sum is cosine cosine sine sine, cosine theta cosine theta minus sine theta sine theta. This becomes cosine squared theta minus sine squared theta and that's our double angle identity for cosine.