Question

If α + β = 90° and $\mathrm{\alpha }=\frac{\mathrm{\beta }}{2}$, then tan α tan β = ________.

Answer 1

$$\begin{gathered} \alpha +\beta =90° \\ \Rightarrow \frac{\beta }{2}+\beta =90°\left[\alpha =\frac{\beta }{2}\right] \\ \Rightarrow \frac{3\beta }{2}=90° \\ \Rightarrow \beta =\frac{90°\times 2}{3}=60° \end{gathered}$$
$\therefore \alpha =\frac{60°}{2}=30°$
Now,
$\tan \alpha \tan \beta =\tan 30°\times \tan 60°=\frac{1}{\sqrt{3}}\times \sqrt{3}=1$

If α + β = 90° and $\mathrm{\alpha }=\frac{\mathrm{\beta }}{2}$, then tan α tan β = ____1____.