I have Signals and Systems course. Generally we use $j$ instead of $i$. So in here you can think $j$ as a complex number $i$.
In the solution which uses Euler's first step, i see that transform.
$2j(e)^{-j2w}\sin(2w)=2\sin(2w)\cdot (e)^{j(\pi/2-2w)}$.
I underlined it in the screenshot. How can we have transform like that?
Think of the intuition behind complex multiplication. Recall that any complex number $|z|=1$ (The distance from the origin to $z$) can be represented on the unit circle by Euler's Theorem as $z=e^{j\theta}$, where $\theta$ is that angle between the ray that runs from the origin to $z$ and ray starting at the origin and running along the x-axis.
By this result, the product of two complex numbers $z_1,\: z_2$ can be expressed as $z_1z_2=e^{j(\theta_1+\theta_2)}$. All we are doing here is rotating the point around the origin.
To complete the idea, note that $j$ corresponds to a ray along the y-axis, and $|j|=1$. Hence, $j=e^{j\pi/2}$. Your result follows.
Source https://en.wikipedia.org/wiki/Euler%27s_formula