You can nearly:
Sine2 + Cosine2 = 1 so cos2 = 1 - sin2
so that cos = sqrt(1 - sin2)
Unfortunately, you cannot tell whether it is the principal sqrt or its negative equivalent.
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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.
No, they do not.
The maximum of the sine and cosine functions is +1, and the minimum is -1.