If 2 tan b + cot b=tan a the prove that cot b= 2 tan( a-b)

2tan(A - B)

2{(tanA - tanB)/(1+tanA.tanB)}

now substitute for tanA = 2tanB + cot B,

2{(2tanB + cotB - tanB)/(1 + (2tanB + cotB).tanB)}

2{(tanB + cotB)/(2 + 2tan^2B)}   (using cotB*tanB = 1)

{(tanB + cotB)/(1 + tan^2B)}

cotB{(tan^2B + 1)/(1 + tan^2B)}   (using cotB*tanB = 1)