Question

If $10{\sin }^4\alpha +15{\cos }^4\alpha -6$ , find the value of $27\cos e{c}^6\alpha +8{\sec }^6\alpha$

Answer 1

Given $10{\sin }^4\alpha +15{\cos }^4\alpha =6$

$\Rightarrow 10{\sin }^4\alpha +15(1-{\sin }^2\alpha {)}^2=6$

$\Rightarrow 10{\sin }^4\alpha +15\alpha (1+{\sin }^4\alpha -2{\sin }^2\alpha )=6$

$\Rightarrow 25{\sin }^4\alpha -30{\sin }^2\alpha +15=0$

$\Rightarrow \sin \alpha =\sqrt{\frac{3}{5}}$

$\csc \alpha =\sqrt{\frac{5}{3}}$

Also $\cos \alpha =\sqrt{\frac{2}{5}}\Rightarrow \alpha =\sqrt{\frac{5}{2}}$

$\therefore 27{\csc }^6\alpha +8{\sec }^6\alpha$

$27{\left[{\left(\frac{5}{3}\right)}^{1/2}\right]}^6+8{\left[{\left(\frac{5}{2}\right)}^{1/2}\right]}^6=\left[{\left(\frac{5}{3}\right)}^3\right]+8{\left(\frac{5}{2}\right)}^3$

$=125+125=250$