The expression `(1 + tan x + tan^2 x)(1-cot x + cot^2 x)` has the positive values for x, given by

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cot x-2 cot 2x = tan x

The value of the expression on (cot x -tan x)/(cot2 x ) is :

tan^(-1)(cot x)+cot^(-1)(tan x)

If tan(cot x) = cot (tan x) , then sin 2x =

tan^(-1) (cot x) +cot^(-1)(tan x) =

tan^(-1) (cot x) +cot^(-1)(tan x) =pi -2x

tan^(-1)(cot x)-tan^(-1)(cot2x)=