Simplify if $$\frac{a}{b} = \frac{4}{5}$$   and $$\frac{b}{c} = \frac{{15}}{{16}},$$   then $$\frac{{{c^2} - {a^2}}}{{{c^2} + {a^2}}}$$   is = ?

$$\eqalign{ & \frac{a}{b} = \frac{4}{5}{\text{ and }}\frac{b}{c} = \frac{{15}}{{16}} \cr & \Rightarrow \left( {\frac{a}{b} \times \frac{b}{c}} \right) = \left( {\frac{4}{5} \times \frac{{15}}{{16}}} \right) \cr & \Rightarrow \frac{a}{c} = \frac{3}{4} \cr & \therefore \,\frac{{{c^2} - {a^2}}}{{{c^2} + {a^2}}} \cr & = \frac{{1 - \left( {\frac{{{a^2}}}{{{c^2}}}} \right)}}{{1 + \left( {\frac{{{a^2}}}{{{c^2}}}} \right)}} \cr & = \frac{{1 - {{\left( {\frac{a}{c}} \right)}^2}}}{{1 + {{\left( {\frac{a}{c}} \right)}^2}}} \cr & = \frac{{1 - \frac{9}{{16}}}}{{1 + \frac{9}{{16}}}} \cr & = \frac{{\left( {\frac{7}{{16}}} \right)}}{{\left( {\frac{{25}}{{16}}} \right)}} \cr & = \frac{7}{{25}} \cr} $$