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If $0< \alpha, \beta< \pi$ and $\cos\alpha + \cos\beta-\cos (\alpha + \beta) =3/2$ then prove $\alpha = \beta= \pi/3$

How do I solve for $\alpha$ and $\beta$ when only one equation is given? By simplification I came up with something like $$ \sin\frac{\alpha}{2} \sin\frac{\beta}{2} \cos \frac {\alpha +\beta}{2}=\frac{1}{8}. $$ I don't know if this helps. How to do this?

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  • $\begingroup$ are $\alpha,\beta$ angles in a given triangle? $\endgroup$ – Dr. Sonnhard Graubner Jan 14 '17 at 15:38
  • $\begingroup$ @Dr.SonnhardGraubner nope $\endgroup$ – ray Jan 14 '17 at 16:07
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Set $p=\cos\beta$ and $q=\sin\beta$; then your equality can be written $$ \cos\alpha+p-p\cos\alpha+q\sin\alpha=\frac{3}{2} $$ or $$ (1-p)\cos\alpha+q\sin\alpha=\frac{3}{2}-p $$ Set $X=\cos\alpha$ and $Y=\sin\alpha$; then the equation becomes $$ \begin{cases} (1-p)X+qY=\frac{3}{2}-p\\[6px] X^2+Y^2=1 \end{cases} $$ and the distance of the line from the origin should be $\le1$, or the line and the circle wouldn't meet: $$ \frac{(\frac{3}{2}-p)^2}{(1-p)^2+q^2}\le1 $$ that becomes $$ \frac{9}{4}-3p+p^2\le 1-2p+p^2+q^2 $$ that is, recalling that $p^2+q^2=1$, $$ p^2-p+\frac{1}{4}\le0 $$ Can you finish up?

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HINT:

Use Prosthaphaeresis Formula on $\cos\alpha,\cos\beta$

and Double angle formula on $$\cos 2\cdot\dfrac{\alpha+\beta}2$$ to get

$$2\cos^2\dfrac{\alpha+\beta}2-2\cos\dfrac{\alpha-\beta}2\cos\dfrac{\alpha+\beta}2+\dfrac12=0\ \ \ \ (1)$$ which is a Quadratic Equation in $\cos\dfrac{\alpha+\beta}2$ whose discriminant must be $\not<0$

i.e., $$\left(2\cos\dfrac{\alpha-\beta}2\right)^2-4=-2\sin^2\dfrac{\alpha-\beta}2\ge0$$

But for real $\dfrac{\alpha-\beta}2,$ $$\sin^2\dfrac{\alpha-\beta}2\ge0$$

So, we have $$\sin^2\dfrac{\alpha-\beta}2=0\iff\sin\dfrac{\alpha-\beta}2=0$$

$\implies\dfrac{\alpha-\beta}2=n\pi$ where $n$ is any integer

But as $0<\alpha,\beta<\pi.n=0\implies\alpha-\beta=0$

So, $(1)$ is reduced to $$0=2\cos^2\beta-2\cos\beta+\dfrac12=\dfrac{(2\cos\beta-1)^2}2$$

So, $\cos\beta=\dfrac12=\cos\dfrac\pi6\implies\beta=2m\pi\pm\dfrac\pi6$ where $m$ is any integer

But as $0<\alpha<\pi$

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  • $\begingroup$ with prosthaphaeresis formula, I get a $\cos (\frac {\alpha - \beta}{2})$. How to deal with it? $\endgroup$ – ray Jan 14 '17 at 16:12
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    $\begingroup$ @ray, Please find the updated answer. $\endgroup$ – lab bhattacharjee Jan 15 '17 at 3:37

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