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  4. If 10 sin 4 α+15 cos 4 α=6, then the value of 81 operatornamecosec8 α+18 sec 8 α is equal to

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Q. If $10 \sin ^4 \alpha+15 \cos ^4 \alpha=6$, then the value of $81 \operatorname{cosec}^8 \alpha+18 \sec ^8 \alpha$ is equal to ___

Trigonometry

Solution:

Given: $10 \sin ^4 \alpha+15 \cos ^4 \alpha=6$
Divide the above equation by 6 on both sides.
$\frac{10 \sin ^4 \alpha}{6}+\frac{15 \cos ^4 \alpha}{6}=\frac{6}{6} $
$ \left(\frac{5}{3} \sin ^2 \alpha\right) \cdot \sin ^2 \alpha+\left(\frac{5}{2} \cos ^2 \alpha\right) \cdot \cos ^2 \alpha=1$
$ \because 10 \sin ^4 \alpha+15 \cos ^4 \alpha=1$
$ \frac{5}{3} \sin ^2 \alpha=1 \text { and } \frac{5}{2} \cos ^2 \alpha=1$
$ \frac{5}{3} \operatorname{cosec}^2 \alpha=1 \text { and } \frac{5}{2} \sec ^2 \alpha=1$
Hence, $81 \operatorname{cosec}^8 \alpha+16 \sec ^8 \alpha=$ ?
$ =81\left(\operatorname{cosec}^2 \alpha\right)^4+16\left(\sec ^{28} \alpha\right)^4 $
$ =81\left(\frac{5}{3}\right)^4+16\left(\frac{5}{2}\right)^4=81 \times \frac{625}{81}+16 \times \frac{625}{16} $
$ =625+625=1250$