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\[\huge y = \cos(i \theta) + i \sin(i \theta)\] prove that y ϵ R
Use the definitions \[\sin(z)=\frac{e ^{iz}-e ^{-iz}}{2i}~~~\cos(z)=\frac{e ^{iz}+e ^{-iz}}{2}\]Substitute i theta for z, and it will be evident that both terms are real numbers.
since : \[\huge e^{i \theta } = \cos(\theta) + isin(\theta)\] therefore: \[\huge \cos(i \theta) + isin(i \theta) = e^{i i \theta} = e^{- \theta} = ({1 \over e})^{\theta} \] and \[\huge e \ and \ \theta \in \mathbb{R} \] so \[\huge y \in \mathbb{R}\]