Question

Prove that :
${\sin }^{4}A+{\cos }^{4}A=1-2{\sin }^{2}A{\cos }^{2}A$.

Answer 1

We, have
LHS = $si{n}^{4}A$ + $co{s}^{4}A$

$\Rightarrow$ LHS = $(si{n}^{2}A{)}^2+(co{s}^{2}A{)}^2+2si{n}^{2}Aco{s}^{2}A-2si{n}^{2}Aco{s}^{2}A$ [Adding and subtracting 2 $si{n}^{2}A$ $co{s}^{2}A$ ]

$\Rightarrow$ LHS = $(si{n}^{2}A+co{s}^{2}A{)}^2-2si{n}^{2}Aco{s}^{2}A=1-2si{n}^{2}Aco{s}^{2}A=RHS$