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Given that
tan2x = cot (x+ π/3)
= cot(x + 60)
= cot(90 + x -30)
= tan(x - 30)
=> tan2x = tan (x- π/6)
=> 2x = n*π + (x- π/6)
=> x = n*π - π/6 (Here n ∈ Z)