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Mathematics
If 15 sin 4 α+10 cos 4 α=6, for some α ∈ R, then the value of 27 sec 6 α+8 operatornamecosec6 α is equal to :
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Q. If $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$, for some $\alpha \in R$, then the value of $27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha$ is equal to :
JEE Main
JEE Main 2021
Trigonometric Functions
A
350
15%
B
500
32%
C
400
17%
D
250
36%
Solution:
$15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$
$15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{2}$
$\left(3 \sin ^{2} \alpha-2 \cos ^{2} \alpha\right)^{2}=0$
$\tan ^{2} \alpha=\frac{2}{3} \cdot \cot ^{2} \alpha=\frac{3}{2}$
$\Rightarrow 27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha$
$=27\left(\sec ^{6} \alpha\right)^{3}+8\left(\operatorname{cosec}^{6} \alpha\right)^{3}$
$=27\left(1+\tan ^{2} \alpha\right) 3+8\left(1+\cot ^{2} \alpha\right)^{3}$
$=250$