A math professor of mine once said of this (paraphrased): yes it is cool, but really it is just a trivial consequence of how we defined our extension of the real exponential map to complex numbers. The really cool part, is that this extension is the _only_ possible holomorphic (differentiable for complex numbers) extension. All that to say that the really nifty part of this identity is the uniqueness of analytic continuations (on the whole complex plane).

It's really just a nice result of euler's theorem:

    e^(a * i) = cos(a) + i * sin(a)

Where `a` is the angle in radians and `i` is `sqrt(-1)`

Well, pi in there is a special case. The general case for any angle $\theta$ is $e^{\i\theta}=\cos(\theta)+\i\sin(\theta)$ It lets you transform between rectangular and polar representations of complex numbers. That allows for things like phasors [1] which are used quite a lot in electrical engineering. It's also used in Fourier analysis, IE all sorts of digital signal processing. Basically anywhere you get complex numbers, sines, or cosines, Euler's formula shows up. That's a lot of places in math/physics/engineering.

[1] https://en.wikipedia.org/wiki/Phasor

In the form you wrote it, there are no real applications. But, written as $e^{i\theta} = \cos \theta + i \sin \theta$, it's frequently used in complex analysis and other areas.