Question:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Solution:
$\mathrm{LHS}=2 \cot (\alpha-\beta)$
$=\frac{2(1+\tan \alpha \tan \beta)}{[\tan \alpha-\tan \beta]}$
$=\frac{2+2(x+1)(x-1)}{(x+1-x+1)}$
$=\frac{2+2 x^{2}-2}{2}$
$=\frac{2 x^{2}}{2}$
$=x^{2}$
= RHS
Hence proved.