What is sin ^2 z+cos ^2 z for any z∈ℂ?

Exercise · ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-003 · ★☆☆ Basic

What is \(\sin^2 z+\cos^2 z\) for any \(z\in\mathbb{C}\)?

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Substituting the exponentials, \(\sin^2z+\cos^2z=\frac{-(e^{2iz}-2+e^{-2iz})+(e^{2iz}+2+e^{-2iz})}{4}=1\); the real identity is preserved.

The sine \(\sin z\) and cosine \(\cos z\) of a complex number \(z\) are the functions that extend the real ones through the complex exponential \(e^{iz}\).

\[\sin z=\frac{e^{iz}-e^{-iz}}{2i},\qquad \cos z=\frac{e^{iz}+e^{-iz}}{2}\]