Question

Prove that :
$(1+\tan A{)}^2+(1-\tan A{)}^2=2{\sec }^{2}A$

Answer 1

Take left hand side of the equation ${\left(1+\tan A\right)}^2+{\left(1-\tan A\right)}^2=2{\sec }^{2}A$.

$LHS$ = ${\left(1+\tan A\right)}^2+{\left(1-\tan A\right)}^2=1+{\tan }^{2}A+2\tan A+1+{\tan }^{2}A-2\tan A$

$=2+2{\tan }^{2}A$

$=2\left(1+{\tan }^{2}A\right)$

$=2{\sec }^{2}A$

$=RHS$