Question

Evaluate :
${sec}^{4}A(1-{sin}^{4}A)-2{tan}^{2}A=1$

Answer 1

${\sec }^{4}A(1-{\sin }^{4}A)-2{\tan }^{2}A=1$

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

$\Rightarrow 1-{\sin }^{4}A-2{\sin }^{2}A{\cos }^{2}A={\cos }^{4}A$

$\Rightarrow 1-2{\sin }^{2}A{\cos }^{2}A={\cos }^{4}A+{\sin }^{4}A$

$\Rightarrow 1={\sin }^{4}A+{\cos }^{4}A+2{\sin }^{2}A{\cos }^{2}A$

$\Rightarrow 1=({\sin }^{2}A+{\cos }^{2}A{)}^2$

$\Rightarrow 1=1^2$

$\Rightarrow 1=1$

Hence proved.