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Sine(A+ B) = Sine(A)*Cosine(B) + Cosine(A)*Sine(B).
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
Tangent = sine/cosine provided that cosine is non-zero. When cosine is 0, then tangent is undefined.
They are different trigonometric functions!
No, they do not.