$\left[\csc^2\frac{\alpha}{2}+\csc^2\frac{\beta}{2}+\csc^2\frac{\gamma}{2}\right]=2$

Question

If $\cos \alpha \cos \beta \cos \gamma-\cos \alpha-\cos \beta-\cos \gamma+1=0$
and $\alpha\neq\beta\neq\gamma\neq2n\pi$,then prove that $\left[\csc^2\frac{\alpha}{2}+\csc^2\frac{\beta}{2}+\csc^2\frac{\gamma}{2}\right]=2$

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$\cos \alpha \cos \beta \cos \gamma+1=\cos \alpha+\cos \beta+\cos \gamma$
$\cos \alpha \cos \beta \cos \gamma+1=2\cos^2 \frac{\alpha}{2}+2\cos^2 \frac{\beta}{2}+2\cos^2 \frac{\gamma}{2}-3$

But there seems no method to calculate the value of the required expression.Some hints and suggestions are needed.

Answer 1

HINT:

As $\csc^2\dfrac x2=\dfrac2{1-\cos x},$

Please simplify $$\dfrac2{1-\cos\alpha}+\dfrac2{1-\cos\beta}+\dfrac2{1-\cos\gamma}=2$$ to reach at the given condition