Question

Prove that cot beta = 2 tan (alpha-beta) if 2tan beta + cot beta = 1.

Answer 1

2tan(α−β) =2[ tanα−tanβ 1+tanα.tanβ ] =2[ 2tanβ+cotβ−tanβ 1+(2tanβ+cotβ).tanβ ] [as,tanα=2tanβ+cotβ] =2[ tanβ+cotβ 1+2 tan 2 β+1 ] = 2(tanβ+cotβ) 2+2 tan 2 β = 2(tanβ+ 1 tanβ ) 2(1+ tan 2 β) = 1 tanβ =cotβ So, cotβ=2tan(α−β)