Question

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

Answer 1

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

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

$={({\sin }^2\theta +{\cos }^2\theta )}^2-4{\sin }^2\theta {\cos }^2\theta$

$={\left(1\right)}^2-4{\sin }^2\theta {\cos }^2\theta$ since ${\sin }^2\theta +{\cos }^2\theta =1$

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

Hence proved.