If $\cos 2B=\frac{\cos \left(A+C\right)}{\cos \left(A-C\right)}$  then $\tan A,\tan B,\tan C$ are in

$\Rightarrow \frac{\cos 2B+1}{\cos 2B-1}=\frac{\cos \left(A+C\right)+\cos \left(A-C\right)}{\cos \left(A+C\right)-\cos \left(A-C\right)}$

$\Rightarrow \frac{2{\cos }^{2}B}{-2{\sin }^{2}B}=\frac{2\cos A.\cos C}{-2\sin A.\sin C}$

$\Rightarrow {\cot }^{2}B=\cot A.\cot C$

$\Rightarrow \cot A,\cot B,\cot C\text{are}\text{in}G.P$

$\left(or\right)\tan A,\tan B,\tan C\text{are}\text{in}G.P$