sin^2(x)+cos^2(x) = 1
tan^2(x) + 1 = sec^2(x) <-- i think this was what you were thinking of in some sense
cot^2(x) + 1 = csc^2(x)
sin^2(x) = (1 - cos2x) / 2
cos^2(x) = (1 + cos2x) / 2
You can use the Euler formula e^ix = cos(x) + i sin(x) to get identities for sin() and cos() in terms of complex exponentials, then the other identities fall out easily.
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