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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)
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)
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