Because x^2 + y^2 = 1 defines a unit circle, if you perform x^2 + i * y^2 = 1, you get a unit circle over the real/imaginary planes.
Or to put it in a picture: https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Eu...
Remember the definition of sin / cos: https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and...
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Once you realize that "the imaginary axis" is just an arbitrary 2-dimension extension field (and that "imaginary" is a very bad name for it), it becomes way easier to see.
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Now if you had a 3-dimension graph (real, imaginary, and time), and then you project the unit-circle over the real + time axis, you get a sin (or cos) respectively.
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Or to put it another way: there's a periodic nature of i:
* i = i
* i^2 = -1
* i^3 = -i
* i^4 = 1 -- Cycle-length 4
* i^5 = i
* i ^ x = i^(x mod 4) (In general)
This period itself forms a circle, moving from real-to-imaginary and back. (e^i^x) = (e^(ix)) therefore is periodic as well. And the most natural periodic cycle is a circle (which is of course, just sin/cos).
To make this "period 4" into a proper circle, multiply by pi/2.
i^(pi/2)^x == period 3.141592.... or the "circumference" of the unit circle.
e^i^(pi/2)^x == e^(i * pi/2 * x). Done.
So you need at least that one, since it is by convention, and not really derivable[1]:
I personally just use Euler's notation (using exp) as a tool to derive trig formulas, since I know power composition rules quite well already, and those tend to be more useful in general. Is just the Pythagorean theorem on a unit circle, which also defines sin and cos :)[1] OK, you can look at the taylor series expansion, but you need to remember the derivatives for this.