If $10{\sin }^{4}\theta +15{\cos }^{4}\theta =6$, then the value of $\frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{16{\sec }^{8}\theta }$ is: [JEE Main 2025, 4 Apr (Shift 1)]

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YESPYQ · Accepted Answer

Correct answer: a) $\frac{2}{5}$. $10{\sin }^{4}\theta +15{\cos }^{4}\theta =6$ $\Rightarrow 10\left[1-2 \sin ^2 \theta \cos ^2 \theta\right]+5 \cos ^4 \theta=6$ $\Rightarrow 25 \cos ^4 \theta-20 \cos ^2 \theta+4=0$ $\Rightarrow\left(5 \cos ^2 \theta-2\right)^2=0$ $\Rightarrow {\cos }^{2}\theta =\frac{2}{5}$ and ${\sin }^{2}\theta =\frac{3}{5}$ $\Rightarrow \frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{16{\sec }^{8}\theta }\ =\frac{27\cdot {\left(\frac{5}{3}\right)}^{3}+8\cdot {\left(\frac{5}{2}\right)}^{3}}{16\cdot {\left(\frac{5}{4}\right)}^{4}}\ =\frac{2}{5}$

If \(10{\sin }^{4}\theta +15{\cos }^{4}\theta =6\), then the value of \(\frac{27{\csc }^{6}\theta +8{\sec }^{6}\theta }{…

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