It appears to me that all trig identities and derivative rules are of the same form for complex numbers as in real numbers. Are there any exceptions?
2026-03-26 05:56:23.1774504583
Derivative rules and trig identities for complex variables
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The following trigonometric identity holds for any real number $x$, $$|\cos x|^2+|\sin x|^2=1.$$ Is it true that $|\cos z|^2+|\sin z|^2=1$ for all $z\in \mathbb{C}$?