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Question Number 18062 by Tinkutara last updated on 14/Jul/17

$$\mathrm{If}0<\alpha ,\beta <\pi \mathrm{and}\mathrm{they}\mathrm{satisfy}$$ $$\cos \alpha +\cos \beta -\cos (\alpha +\beta )=\frac{3}{2}$$ $$\left(1\right)\alpha =\beta$$ $$\left(2\right)\alpha +\beta =\frac{2\pi }{3}$$ $$\left(3\right)\alpha =2\beta$$ $$\left(4\right)\beta =2\alpha$$

Answered by Tinkutara last updated on 16/Jul/17

$$2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}-2{\cos }^2\frac{\alpha +\beta }{2}=\frac{1}{2}$$ $$4{\cos }^2\frac{\alpha +\beta }{2}-4\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}+1=0$$ $${(2\cos \frac{\alpha +\beta }{2}-\cos \frac{\alpha -\beta }{2})}^2+{\sin }^2\frac{\alpha -\beta }{2}=0$$ $$\Rightarrow \alpha =\beta =\frac{\pi }{3}$$

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