Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
The equation $\cos x\cos 6x=-1$:
has 50 solutions in [0,100π]
has 3 solutions in [0,3π]
has even number of solutions in (3π,13π)
has one solution in [π/2,π]
Step-by-Step Solution
Key Concept: The equation $AB = -1$ with $|A|, |B| ≤ 1$ forces both factors to achieve extreme values simultaneously.
From $\cos x\cos 6x = -1$, we need both $\cos x = 1$ and $\cos 6x = -1$ simultaneously, or $\cos x = -1$ and $\cos 6x = 1$. Case 1 is impossible. Case 2 requires $\cos x = -1$ and $\cos x = -1$, giving $x = (2n+1)\pi$ for $n \in \mathbb{Z}$.
Correct Answer: 1,3,4