(related to Proposition: Pythagorean Identity)
From the Euler's formula and the definition of the unit circle, it follows
\[|\cos(x)+i\sin(x)|^2=|\exp(ix)|^2=1\quad\quad x\in\mathbb R.\]
From the definition of the absolute value for complex numbers, it follows
\[\left(\sqrt{\cos^2(x)+\sin^2(x)}\right)^2=1\quad\quad x\in\mathbb R.\]
From the properties of \(n\)-th roots, it follows
\[\cos^2(x)+\sin^2(x)=1\quad\quad x\in\mathbb R.\]
