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QUESTION From the formulas for sin and cos in terms of the
exponential function, prove that$\sin^2z+\cos^2 z=1$, for all
complex numbers $z$.



ANSWER
$\sin^2z+\cos^2z=(\frac{e^{iz}-e^{-iz}}{2i})^2+(\frac{e^{iz}+e^{-iz}}{2})^2=1$
after expanding.

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