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I am lacking in understanding in the cosine and sine angle addition formulas. I have seen several questions similar to this but I have not seen an answer that explains how this conclusion can be derived. Geometric proofs offer little intuition and the rotation matrix is derived from the cosine and sin angle addition formulas. I haven't been able to derive this from calculus either and am already struggling on figuring out the sine and cosine derivatives in a way that makes sense instead of blindly accepting it. I would appreciate it if someone could answer why the cosine and sine angle addition formulas are as they are in an intuitive manner.