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Question Number 48225 by gunawan last updated on 21/Nov/18

$$\mathrm{prove}\mathrm{that}$$$$\exp \left(\frac{2+\pi \mathrm{i}}{4}\right)=\sqrt{\frac{e}{2}}(1+i)$$$$\cos (z_1+z_2)=\cos z_1\cos z_2-\sin z_1\sin z_2$$

Answered by Smail last updated on 21/Nov/18

$$e^{\frac{2+\pi i}{4}}=e^{\frac{1}{2}+i\frac{\pi }{4}}=\sqrt{e}\times e^{i\pi /4}=\sqrt{e}\times (cos\left(\frac{\pi }{4}\right)+isin\left(\frac{\pi }{4}\right))$$$$=\sqrt{e}(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})=\frac{\sqrt{e}}{\sqrt{2}}(1+i)$$$$=\sqrt{\frac{e}{2}}(1+i)$$

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