Question

Expression $\frac{{\sin }^6\theta -{\cos }^6\theta }{{\sin }^2\theta -{\cos }^2\theta }$ is equal to

• A. ${\sin }^4\theta -{\cos }^4\theta$
• B. $1-{\sin }^2\theta {\cos }^2\theta$
• C. $1+{\sin }^2\theta {\cos }^2\theta$
• D. $1-3{\sin }^2\theta {\cos }^2\theta$

Answer 1

The correct option is B $1-{\sin }^2\theta {\cos }^2\theta$

$\frac{{\sin }^6\theta -{\cos }^6\theta }{{\sin }^2\theta -{\cos }^2\theta }=\frac{({\sin }^2\theta {)}^3-({\cos }^2\theta {)}^3}{{\sin }^2\theta -{\cos }^2\theta }$
$=\frac{({\sin }^2\theta -{\cos }^2\theta )({\sin }^4\theta +{\cos }^4\theta +{\sin }^2\theta {\cos }^2\theta )}{({\sin }^2\theta -{\cos }^2\theta )}$
$=\left[\right({\sin }^2\theta +{\cos }^2\theta {)}^2-2{\sin }^2\theta \times {\cos }^2\theta +{\sin }^2\theta {\cos }^2\theta ]$
$=1-{\sin }^2\theta {\cos }^2\theta$.