$$\begin{align} \sinh(x)&=-i\sin(ix) \\ \cosh(x)&=\phantom{-i}\cos(ix) \end{align}$$
Why are these identities true?
Other than using algebra and formulas, is there a more intuitive/geometric way to show that the above is true?
Attempt
The equation of a unit circle is: $x^2+y^2=1$
The equation of a unit hyperbola is: $x^2-y^2=1$
If $y\rightarrow iy$, then a circle will become a hyperbola.
For a circle: $y=\sin(θ)$
For a hyperbola: $iy=\sinh(θ)$ ---> $y=-i\sinh(θ)$
For a circle: $x=\cos(θ)$
For a hyperbola: $x=\cosh(θ)$
But obviously $\cos(x)=\cosh(x)$ and $\sin(x)=-i\sinh(x)$ are wrong.