> 
> Given:
> 
> 
> $$\cot (A-B) = 3 \qquad \cot (A+B) = 2$$
> 
> 
> Evaluate:
> 
> 
> $$\tan A + \frac{1}{\cot^2 A} + \tan^3 A + \frac{1}{\cot^4 A} + \cdots $$
> 
> 
> 

Attempts:

I think that the given information is used to get the value of $A$ so I can input $A$ to the problems and using the formula of infinite sum of geometric sequence. I haven't touched that step yet. I am still trying to get the value of $A$.

From
$$\cot (A-B) = \frac{\cot A \cot B-1}{\cot A + \cot B}$$
I get to $\cot A - 5\cot B = 2$.

Now, I am stuck. Where should I do next?