Question

Prove that :
$\frac{si{n}^{2}A}{co{s}^{2}A}+\frac{co{s}^{2}A}{si{n}^{2}A}=\frac{1}{si{n}^{2}Aco{s}^{2}A}-2$

Answer 1

We have,
L.H.S. = $\frac{si{n}^{2}A}{co{s}^{2}A}+\frac{co{s}^{2}A}{si{n}^{2}A}=\frac{si{n}^{4}A+co{s}^{4}A}{si{n}^{2}Aco{s}^{2}A}$

$\Rightarrow$ L.H.S. = $\frac{(si{n}^{2}A{)}^2+(co{s}^{2}A{)}^2+2si{n}^{2}Aco{s}^{2}A-2si{n}^{2}Aco{s}^{2}A}{si{n}^{2}Aco{s}^{2}A}$

$\Rightarrow$ L.H.S. = $\frac{(si{n}^{2}A+co{s}^{2}A{)}^2-2si{n}^{2}Aco{s}^{2}A}{si{n}^{2}Aco{s}^{2}A}$

$\Rightarrow$ L.H.S. = $\frac{1-2si{n}^{2}Aco{s}^{2}A}{si{n}^{2}Aco{s}^{2}A}$

$\Rightarrow$ L.H.S. = $\frac{1}{si{n}^{2}Aco{s}^{2}A}-\frac{2si{n}^{2}Aco{s}^{2}A}{si{n}^{2}Aco{s}^{2}A}=\frac{1}{si{n}^{2}Aco{s}^{2}A}-2=R.H.S.$