If $\frac{\cos (\alpha +\gamma )}{\alpha -\gamma }=\cos 2\beta$ then $\tan \alpha ,\tan \beta and\tan \gamma$ are in ___

$\tan \alpha ,\tan \beta ,\tan \gamma$

$\frac{\cos (\alpha +\gamma )}{\cos (\alpha -\gamma )}=\frac{\cos 2\beta }{1}$

$\frac{\cos (\alpha +\gamma )+\cos (\alpha -\gamma )}{\cos (\alpha +\gamma )-\cos (\alpha -\gamma )}=\frac{\cos 2\beta +1}{\cos 2\beta -1}$

$\Rightarrow \frac{2\cos \alpha \cos \gamma }{-2\sin \gamma \sin \gamma }=\frac{2{\cos }^2\beta }{-2{\sin }^2\beta }$

$\Rightarrow \frac{1}{\tan \alpha \tan \gamma }=\frac{1}{{\tan }^2\beta }$

$\Rightarrow {\tan }^2\beta =\tan \alpha \tan \gamma$

$\tan \alpha \tan \beta ,\tan \gamma$ are in G.P.