The tangent of the angle between the lines whose intercepts on the axes are respectively $a,-b$ and $b,-a$ is

Equations of lines:

$ \frac{x}{a}-\frac{y}{b}=1 $

$ \frac{x}{b}-\frac{y}{a}=1 $

Slopes:

$ m_1=\frac{b}{a},\quad m_2=\frac{a}{b} $

Angle between lines:

$ \tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right| $

$ =\left|\frac{\frac{b}{a}-\frac{a}{b}}{1+\frac{b}{a}\cdot\frac{a}{b}}\right| $

$ =\left|\frac{\frac{b^2-a^2}{ab}}{2}\right| $

$ =\left|\frac{b^2-a^2}{2ab}\right| $

$ \therefore \tan\theta=\pm\frac{b^2-a^2}{2ab} $