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Question Number 75655 by mr W last updated on 15/Dec/19

$$ifz\in \mathbb{C}$$$$is{\sin }^{2}z+{\cos }^{2}z=1alsovalid?$$

Answered by MJS last updated on 15/Dec/19

$$\mathrm{yes}.$$$${\sin }^{2}z={\left(\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}}\right)}^2=\frac{{\mathrm{e}}^{2\mathrm{i}z}-2+{\mathrm{e}}^{-2\mathrm{i}z}}{-4}$$$${\cos }^{2}z={\left(\frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2}\right)}^2=\frac{{\mathrm{e}}^{2\mathrm{i}z}+2+{\mathrm{e}}^{-2\mathrm{i}z}}{4}$$$$\frac{{\mathrm{e}}^{2\mathrm{i}z}-2+{\mathrm{e}}^{-2\mathrm{i}z}}{-4}+\frac{{\mathrm{e}}^{2\mathrm{i}z}+2+{\mathrm{e}}^{-2\mathrm{i}z}}{4}=1$$

Commented by mr W last updated on 15/Dec/19

$$thankyouforcomfirming,sir.$$

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