Question

If $6{\sin }^4\theta +3{\cos }^4\theta =2,$ where $\theta \in R,$ then the value of $8{\sec }^6\theta +3{\text{cosec}}^6\theta$ is equal to

• A. $128$
• B. $256$
• C. $116$
• D. $108$

Answer 1

The correct option is D $108$

Given, $6{\sin }^4\theta +3{\cos }^4\theta =2$
$\Rightarrow 6{\sin }^4\theta +3(1-{\sin }^2\theta {)}^2=2$
$\Rightarrow 6{\sin }^4\theta +3+3{\sin }^4\theta -6{\sin }^2\theta =2$
$\Rightarrow 9{\sin }^4\theta -6{\sin }^2\theta +1=0$
$\Rightarrow (3{\sin }^2\theta -1{)}^2=0$
$\Rightarrow {\sin }^2\theta =\frac{1}{3}$ and ${\cos }^2\theta =\frac{2}{3}$

$\therefore 8{\sec }^6\theta +3{\text{cosec}}^6\theta =8{\left(\frac{3}{2}\right)}^3+3(3{)}^3=108$