Step 1:

the equation of the given lines are :

$ \large\frac{x}{a}$$+\large\frac{y}{b}$$=1$-----(1)

$ \large\frac{x}{a}$$-\large\frac{y}{b}$$=1$--------(2)

This can be written as

$ bx+ay=ab$

or $y=-\large\frac{bx}{a}$$+b$----------(3)

$bx-ay=ab$

or $ y = \large\frac{b}{a}$$x-b$--------(4)

Hence the slopes of the lines (1) and (2) are $ -\large\frac{b}{a}$ and $ \large\frac{b}{a}$ respectively.

Step 2 :

The angle between the lines is

$ \theta = \tan^{-1} \bigg| \large\frac{m_1-m_2}{1+m_1m_2} \bigg|$

Substituting for $m_1$ and $m_2$ we get,

$ \theta = \tan^{-1} \bigg| \large\frac{ -\Large\frac{b}{a}-\Large\frac{b}{a}}{1+\Large\frac{b}{a} \times \large\frac{b}{a}} \bigg|$

$ = \tan^{-1} \bigg| \large\frac{-2\Large\frac{b}{a}}{\Large\frac{a^2+b^2}{a^2}} \bigg|$

$ \therefore \tan \theta = \large\frac{2ab}{a^2+b^2}$

Hence proved.