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Question Number 111939 by Khanacademy last updated on 05/Sep/20

Answered by bemath last updated on 05/Sep/20

$$consider\tan x+\tan y=4$$$$\Rightarrow \frac{\sin x}{\cos x}+\frac{\sin y}{\cos y}=4$$$$\Rightarrow \frac{\sin (x+y)}{\cos x.\cos y}=4\Rightarrow \sin (x+y)=4\times \frac{3}{2}$$$$\sin (x+y)=6.Ifx,y\in \mathbb{R},itis$$$$impossible$$

Commented by aleks041103 last updated on 05/Sep/20

$$Sincesin(x+y)=6,then:$$$$cos(x+y)=\pm \sqrt{1-6^2}=\pm \sqrt{-35}=\pm i\sqrt{35}$$$$Then:$$$$tan(x+y)=\frac{sin(x+y)}{cos(x+y)}=\frac{6}{\pm i\sqrt{35}}=\mp i\sqrt{\frac{36}{35}}$$$$tan(x+y)=\pm i\sqrt{\frac{36}{35}}$$

Answered by $@y@m last updated on 05/Sep/20

$$0⩽\cos x⩽1$$$$0⩽\cos y⩽1$$$$\Rightarrow 0⩽\cos x.\cos y⩽1$$$$So,Pleasecheckthequestion.$$

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