I have tried to post an answer but the site froze up and I lost it all.
Here is the short version
solve tan 2x = - cot (x + pi/3)
I solved by graphing.
Herer is the full graph
https://www.desmos.com/calculator/mblxlza8ms
From the graph it appears the answer is \(x=n\pi+\frac{5\pi}{6}\qquad n \in Z\)
In my last (lost) answer I substituted this answer in to check it was correct and it was exactly so.
If you want to ask questions by all means do so :)
Here is the pic: