Given,

Lines $L_1\equiv \frac{x}{a}+\frac{y}{b}=1$ and $L_2\equiv \frac{x}{b}+\frac{y}{a}=1$,

Now slope of line $L_1$ is $\frac{-b}{a}$ and slope of line $L_2$ is $\frac{-a}{b}$

Now we know that angle between two line is given by,

$\tan \theta =\left|\frac{m_1-m_2}{1-m_1{m}_2}\right|$

$\Rightarrow \tan \theta =\left|\frac{{\displaystyle \frac{b}{a}}-{\displaystyle \frac{a}{b}}}{1+1}\right|$

$\Rightarrow \tan \theta =\left|\frac{{\displaystyle b^2-a^2}}{2ab}\right|$

Now $\sin \theta =\frac{\left|b^2-a^2\right|}{\sqrt{{\left(b^2-a^2\right)}^2+{\left(2ab\right)}^2}}$

$\Rightarrow \sin \theta =\frac{\left|a^2-b^2\right|}{\left|a^2+b^2\right|}$