Question

Prove:

$\frac{{\sin }^{2}A}{{\cos }^{2}A}+\frac{{\cos }^{2}a}{{\sin }^{2}A}=\frac{1}{{\sin }^{2}A{\cos }^{2}A}-2$

Answer 1

$\frac{{sin}^{2}A}{{cos}^{2}A}+\frac{{cos}^{2}A}{{sin}^{2}A}$

$=\frac{{sin}^{4}A+{cos}^{4}A}{{sin}^{2}A{cos}^{2}A}$

$=\frac{{\left({sin}^{2}A\right)}^2+{\left({cos}^{2}A\right)}^2}{{sin}^{2}A{cos}^{2}A}$

$={({sin}^{2}A+{cos}^{2}A)}^2-\frac{2{sin}^{2}A{cos}^{2}A}{{sin}^{2}A{cos}^{2}A}$

$=1-\frac{2{sin}^{2}A{cos}^{2}A}{{sin}^{2}A{cos}^{2}A}$

$=\frac{1}{{sin}^{2}A{cos}^{2}A}-2$