Question

The possible value of ${\sin }^6(\theta )+{\cos }^6(\theta )-3{\cos }^4(\theta )$ is

• A. 2
• B. -2
• C. -3
• D. 3

Answer 1

Answer: (B)

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