Cot x sec4x = cot x + 2 tan x + tan3x
is this equation or identity?

It is an equation:
here is the proof :
The key to this one is that sec^2(x) = 1 + tan^2(x).
So the left side is 
cot(x) (1 + tan^2(x)) (1 + tan^2(x)) 
Expanding this term by term you get 
cot(x) + 2 cot(x) tan^2(x) + cot(x) tan^4(x), 
but since cot(x) tan(x) = 1, that turns into 
cot(x) + 2 tan(x) + tan^3(x), 
which is the same as the right side.