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Question Number 20366 by ajfour last updated on 26/Aug/17

$$\tan {}^{2}x+2\tan x(\sin y+\cos y)+2=0$$$$Findx,y.$$

Answered by mrW1 last updated on 26/Aug/17

$$\mathrm{D}=4{(\sin \mathrm{y}+\cos \mathrm{y})}^2-4\times 2⩾0$$$${(\sin \mathrm{y}+\cos \mathrm{y})}^2⩾2$$$$2{(\sin \mathrm{y}\cos \frac{\pi }{4}+\sin \frac{\pi }{4}\cos \mathrm{y})}^2⩾2$$$${\sin }^2(\mathrm{y}+\frac{\pi }{4})⩾1$$$$\Rightarrow \sin (\mathrm{y}+\frac{\pi }{4})=\pm 1$$$$\Rightarrow \mathrm{y}+\frac{\pi }{4}=2\mathrm{n}\pi \pm \frac{\pi }{2}$$$$\Rightarrow \mathrm{y}=2\mathrm{n}\pi +\frac{\pi }{4}\mathrm{or}2\mathrm{n}\pi -\frac{3\pi }{4}$$$$$$$$\mathrm{with}\mathrm{D}=0$$$$\tan \mathrm{x}=\frac{-2}{2}=-1$$$$\Rightarrow \mathrm{x}=\mathrm{m}\pi -\frac{\pi }{4}$$

Commented by ajfour last updated on 26/Aug/17

$$thanksalot,sir!$$

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