\[\left( \frac{8}{15} \right)^{2} - \left( \frac{4}{5} \right)^{2} = \left( \frac{8}{15} \right)^{2} - \left( \frac{12}{15} \right)^{2} =\]
\[= \left( \frac{8}{15} - \frac{12}{15} \right)\left( \frac{8}{15} + \frac{12}{15} \right) =\]
\[= - \frac{4}{15} \cdot \frac{20}{15} = - \frac{4}{15} \cdot \frac{4}{3} = - \frac{16}{45}.\]