Question

Prove that $\frac{(1-2{\sin }^{2}A)}{{\cos }^{4}A-{\sin }^{4}A}=2{\cos }^{2}A-1$

Answer 1

According to the problem:

L.H.S$=\frac{{\left(1-2si{n}^{2}A\right)}^2}{{\cos }^{4}A-{\sin }^{4}A}$

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

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

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

$={\cos }^{2}A-\left(1-{\cos }^{2}A\right)$

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

Hence proved.