Question

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:

• A. $\frac{2}{5}$
• B. $\frac{3}{4}$
• C. $\frac{3}{5}$
• D. $\frac{1}{5}$

Hint:

Rewrite trigonometric expressions in terms of $\sin^2 \theta$ and $\cos^2 \theta$ to simplify calculations.

Answer 1

The Correct Option is A

1. Rewrite the given equation: \[ 10 \sin^4 \theta + 15 \cos^4 \theta = 6 \] \[ 10 (\sin^2 \theta)^2 + 15 (1 - \sin^2 \theta)^2 = 6 \]
2. Let $u = \sin^2 \theta$: \[ 10u^2 + 15(1 - u)^2 = 6 \] \[ 10u^2 + 15(1 - 2u + u^2) = 6 \] \[ 10u^2 + 15 - 30u + 15u^2 = 6 \] \[ 25u^2 - 30u + 9 = 0 \]
3. Solve the quadratic equation: \[ u = \frac{30 \pm \sqrt{900 - 900}}{50} = \frac{30 \pm 0}{50} = \frac{3}{5} \] \[ \sin^2 \theta = \frac{3}{5}, \quad \cos^2 \theta = \frac{2}{5} \]
4. Calculate the given expression: \[ \frac{27 \csc^6 \theta + 8 \sec^6 \theta}{16 \sec^8 \theta} = \frac{27 \left( \frac{5}{3} \right)^3 + 8 \left( \frac{5}{2} \right)^3}{16 \left( \frac{5}{2} \right)^4} \] \[ = \frac{27 \cdot \frac{125}{27} + 8 \cdot \frac{125}{8}}{16 \cdot \frac{625}{16}} = \frac{125 + 125}{625} = \frac{250}{625} = \frac{2}{5} \] Therefore, the correct answer is (1) $\frac{2}{5}$.