If 10 sin 4 + 15 cos 4 = 6 , then the value of 27 cosec 6 + 8 sec 6 16 sec 8 is [2025]

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Solution

Q.
If $10 \sin^{4} θ + 15 \cos^{4} θ = 6$ , then the value of $\frac{27 {cosec}^{6} θ + 8 \sec^{6} θ}{16 \sec^{8} θ}$ is [2025]

1 $\frac{3}{4}$

2 $\frac{2}{5}$

3 $\frac{3}{5}$

4 $\frac{1}{5}$

Ans.
(2)

We have, $10 \sin^{4} θ + 15 \cos^{4} θ = 6$

$\Rightarrow 10 \sin^{4} θ + 15 + 15 \sin^{4} θ - 30 \sin^{2} θ = 6$

Let $\sin^{2} θ = p$

$\Rightarrow 10 p^{2} + 15 + 15 p^{2} - 30 p = 6$

$\Rightarrow 25 p^{2} - 30 p = - 9 \Rightarrow {(5 p - 3)}^{2} = 0$

$∴ \sin^{2} θ = 3 / 5 and \cos^{2} θ = 2 / 5$

So, $\frac{27 {cosec}^{6} θ + 8 \sec^{6} θ}{16 \sec^{8} θ} = \frac{27 \times \frac{125}{27} + 8 \times \frac{125}{8}}{16 \times \frac{625}{16}}$

$= \frac{125 + 125}{625} = \frac{2}{5}$ .

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