We need some restrictions here. I'm presuming that x is a real number so we have to be careful about \(\displaystyle cos( x^2 ) \leq 0\) for various values of x.

I do not understand!I think that \(\displaystyle \left(e^{i x^2}\right)^{x} =e^{i^x\cdot x^{2x}}\) where \(\displaystyle x\in \mathbb R , x>0\) is an identity and so \(\displaystyle \left(e^{i x^2}\right)^{x} = e^{i x^3}\) is an equation.
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How do we calculate \(\displaystyle [\cos(x^2)+i \sin(x^2)]^x\)?