Sorry for not replying as I lost my details due to a computer crash.
Z is not a complex variable and yes I do know the definition of the complex function cot Z.
The above identity was from an exam question for A-Levels passed to me and the full context of the question was
tan z + 2 tan2z + 4 tan4z +8 tan8z + 16tan16z=cot z -32 cot 32 z
I only posted part of the question which included a hint from someone: to work on right hand side by integrating it and expanding it to - log cos16 z x cos 8 z x cos 4z x co2z x cos z and finally differentiating it.
The differentiation part is easy and immediately results in the left hand side answer, no problem.
Integrating results in a ln sinz etc , but the expansion is not so easy.