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Question Number 184622 by Shrinava last updated on 09/Jan/23

$$\mathrm{Solve}\mathrm{for}\mathrm{real}\mathrm{numbers}:$$$$\mathrm{sinx}\sqrt{1-{\sin }^2\mathrm{x}}=1+\mathrm{cosy}\sqrt{1-{\cos }^2\mathrm{y}}$$

Answered by floor(10²Eta[1]) last updated on 09/Jan/23

$$\mathrm{sinxcosx}=1+\mathrm{senycosy}$$$$\frac{\mathrm{sen}\left(2\mathrm{x}\right)}{2}=1+\frac{\mathrm{sen}\left(2\mathrm{y}\right)}{2}$$$$\mathrm{sen}\left(2\mathrm{x}\right)=2+\mathrm{sen}\left(2\mathrm{y}\right)$$$$-1⩽\mathrm{sen}\left(2\mathrm{x}\right)⩽1$$$$1⩽2+\mathrm{sen}\left(2\mathrm{y}\right)⩽3$$$$\Rightarrow \mathrm{sen}2\mathrm{x}=1=2+\mathrm{sen}2\mathrm{y}$$$$\Rightarrow \mathrm{sen}2\mathrm{x}=1\wedge \mathrm{sen}2\mathrm{y}=-1$$$$2\mathrm{x}=\frac{\pi }{2}+2\mathrm{n}\pi \wedge 2\mathrm{y}=\frac{3\pi }{2}+2\mathrm{m}\pi$$$$\Rightarrow \mathrm{x}=\frac{\pi }{4}+\mathrm{n}\pi \wedge \mathrm{y}=\frac{3\pi }{4}+\mathrm{m}\pi ,\mathrm{m},\mathrm{n}\in \mathbb{Z}$$

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