Prove the following

Question:

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

  1. (1) 350

  2. (2) 500

  3. (3) 400

  4. (4) 250


Correct Option: , 4

Solution:

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

$15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{2}$

$\left(3 \sin ^{2} \alpha-2 \cos ^{2} \alpha\right)^{2}=0$

$\tan ^{2} \alpha=\frac{2}{3} \cdot \cot ^{2} \alpha=\frac{3}{2}$

$\Rightarrow 27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha$

$=27\left(\sec ^{6} \alpha\right)^{3}+8\left(\operatorname{cosec}^{6} \alpha\right)^{3}$

$=27\left(1+\tan ^{2} \alpha\right) 3+8\left(1+\cot ^{2} \alpha\right)^{3}$

$=250$

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