Question

If $15{\sin }^4\alpha +10{\cos }^4\alpha =6$, for some $\alpha \in R$, then the value of $27{\sec }^6\alpha +8{\mathrm{cosec}}^6\alpha$ is equal to :

• A. 350
• B. 500
• C. 400
• D. 250

Answer 1

Answer: (D)

$15{\sin }^4\alpha +10{\cos }^4\alpha =6$
$15{\sin }^4\alpha +10{\cos }^4\alpha =6{\left({\sin }^2\alpha +{\cos }^2\alpha \right)}^2$
${\left(3{\sin }^2\alpha -2{\cos }^2\alpha \right)}^2=0$
${\tan }^2\alpha =\frac{2}{3}\cdot {\cot }^2\alpha =\frac{3}{2}$
$\Rightarrow 27{\sec }^6\alpha +8{\mathrm{cosec}}^6\alpha$
$=27{\left({\sec }^6\alpha \right)}^3+8{\left({\mathrm{cosec}}^6\alpha \right)}^3$
$=27\left(1+{\tan }^2\alpha \right)3+8{\left(1+{\cot }^2\alpha \right)}^3$
$=250$