Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
If $10\sin^4 u + 15\cos^4 u = 6$ and the value of $9\cos\sec^4 u + 8\sec^4 u$ is $S$, then find the value of $\frac{S}{25}$
Step-by-Step Solution
Key Concept: Expand the quadratic expression and recognize that a perfect square equals zero to find $\cos 2\alpha$.
From $10(1 - \cos 2\alpha) + 15(1 + \cos 2\alpha)^2 = 24$, we expand and simplify to get $(5\cos 2\alpha + 1)^2 = 0$, giving $\cos 2\alpha = -\frac{1}{5}$. Using the identity $\tan^2 \alpha = \frac{1 - \cos 2\alpha}{1 + \cos 2\alpha}$, we obtain $\tan^2 \alpha = \frac{3}{2}$.
Correct Answer: 3