Question

If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x².

Answer 1

$$\begin{gathered} \mathrm{LHS}\hspace{0.17em}=2\cot (\alpha -\beta ) \\ =\frac{2(1+\tan \alpha \tan \beta )}{\left[\tan \alpha -\tan \beta \right]} \end{gathered}$$

$$\begin{gathered} =\frac{2+2\left(x+1\right)\left(x-1\right)}{\left(x+1-x+1\right)} \\ =\frac{2+2x^2-2}{2} \\ =\frac{2x^2}{2} \\ =x^2 \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$