While playing around, I discovered something and am not sure if it is correct.
Assuming that
$e^{it} = \cos(t) + i\sin(t)$
Then $e^i=\cos 1+i\sin 1$
But then
$e^{it} = {(e^i)}^t=\big( \cos 1+i\sin 1 \big)^t$
I have a calculator that unfortunately can only compute to $z^3$ but it was correct for the $2$ values I checked. Is it correct? Does it hold for fractional, negative or complex $t$? I apologize for the title but I do not know how to describe this. Thank you for your patience.
On
By De Moivre's formula for $t$ integer the equality
$$e^{it} = {(e^i)}^t=\big( \cos 1+i\sin 1 \big)^t= \cos (1\cdot t)+i\sin (1\cdot t) $$
holds but for general $t$ it doesn't hold, for that refer to Generalized De Moivre's formula.
The equality$$e^{it}=(\cos1+i\sin 1)^t$$is neither correct nor wrong unless you tell us how do you define $(\cos1+i\sin 1)^t$. Of course, if $t$ is a natural number then this is just $\cos1+i\sin 1$ times itself $t$ times. But how do you define, say, $(\cos1+i\sin 1)^\pi$? Without an answer to this question, your question cannot be answered for a general $t$.