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QUESTION From the definitions of $\sin z$ and $\cos z$ in terms of
the exponential function show that $\frac{d}{dz}(\sin z)=\cos z$
and $\frac{d}{dz}(\cos z)=-\sin z$.


ANSWER $\frac{d}{dz}(\sin
z)=\frac{d}{dz}\frac{e^{iz}-e^{-iz}}{2i}=\frac{i(e^{iz}+e^{-iz})}{2i}=\frac{e^{iz}+e^{-iz}}{2}=\cos
z$ and similarly $\frac{d}{dz}(\cos z)=-\sin z$.


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