Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}$

Question

Question: Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}?$

For instance, $\cos^2(z)+\sin^2(z)=1$ is still true when we move to complex plane but are there identities that are no longer true when we in complex plane?

What about hyperbolic identities? The most identities I encountered seem to be still true in complex.

Many thanks in advance!

Answer 1

If an identity of analytic functions holds on a set with a limit point, then it holds everywhere. Of course $\mathbb R$ is a set with a limit point.

So a counterexample must involve non-analytic functions. Like this

$$ x^2 = |x|^2 $$ holds on $\mathbb R$, but $$ z^2 = |z|^2 $$ fails on most of $\mathbb C$.

Another example, deadly to calculus students: $$ \int\frac{dx}{x}=\log|x|+C $$ true in real calculus, but $$ \int \frac{dz}{z} = \log|z|+C $$ is false in complex calculus. The remedy: $$ \int\frac{dx}{x}=\log x +C $$ is true even in the case of a real variable, but may have a complex constant $C$ since $\log x$ may be complex when $x<0$.