If $\tan A=\frac{2}{3}$, then what is the value of the following?

$\left(5 \sin ^2 A-2 \cos ^2 A\right) \div\left(15 \sin ^2 A+3 \cos ^2 A\right)$

$\frac{2}{87} $

$\left(5 \sin ^2 A-2 \cos ^2 A\right) \div\left(15 \sin ^2 A+3 \cos ^2 A\right)$

Dividing denominator and numerator by cos²A

= $\frac{5\frac{sin^2A}{cos^2A}-2}{15\frac{sin^2A}{cos^2A}+3}$

= $\frac{5tan^2A - 2}{15tan^2A +3}$

Putting the value of tanA

= $\frac{5.\frac{4}{9} - 2}{15.\frac{4}{9} + 3}$

= $\frac{\frac{2}{9}}{\frac{29}{3}}$

= $\frac{2}{87}$