# $i\sin(x)$ phase shift

By Euler's formula, I can express $i$ in the following way: $$i = \cos(\pi/2) + i \sin(\pi/2) = \exp(i \pi/2)$$ I wonder if it is legitimate to write

\begin{align}i \sin x &= \exp(i\pi/2) \cdot \frac{\exp(ix)-\exp(-ix)}{2i} \\ &= \frac{\exp(ix)\exp(i\pi/2)-\exp(-ix)\exp(i\pi/2)}{2i} \\ &= \frac{\exp(i(x+\pi/2))-\exp(-i(x+\pi/2))}{2i} \\ &= \sin(x+\pi/2) \end{align}

I don't feel like this is right, because it would imply a lot of weird things. So where is my mistake? note: I'm really missing sleep, so please be kind with me.

In your sleep-deprived delirium, you have said $$e^{-ix}e^{i\pi/2} = e^{-i(x+\pi/2)}$$ where it is actually $$e^{-ix}e^{i\pi/2} = e^{-i(x-\pi/2)}.$$
It would certainly be strange if $i\sin(x) = \sin(x+\pi/2)$ since one is pure imaginary and the other real for real $x.$