Question

Prove: ${\sin }^4\theta -{\cos }^4\theta =1-2{\cos }^2\theta$

Answer 1

LHS: ${\sin }^4\theta -{\cos }^4\theta$

Using, $a^2-b^2=(a-b)(a+b)$

$=({\sin }^2\theta -{\cos }^2\theta )({\sin }^2\theta +{\cos }^2\theta )$

$=({\sin }^2\theta -{\cos }^2\theta )\left(1\right)$ [$\because {\sin }^2\theta +{\cos }^2\theta =1$]

$=(1-{\cos }^2\theta )-{\cos }^2\theta$ [Using same identity]

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

$=$ RHS

Hence, proved.