Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 12
Question:
Find number of solutions of the equation $\sin^{-1}(\log_2(\cos x)) - 1) + \cos^{-1}(3\log_2^2(\cos x) - 7)) = \frac{\pi}{2}$, if $x \in [0, 4\pi]$.
Step-by-Step Solution
Key Concept: Equate the arguments inside inverse cosine functions and solve the resulting absolute value equation.
Given $\cos^{-1}(3\log_5^2(\cos x)-7)) = \cos^{-1}(|\log_5^2(\cos x)-1|)$, we equate arguments: $|3t-7| = |t-1|$ where $t = \log_5^2(\cos x)$. This gives $3t-7 = t-1$ or $3t-7 = -(t-1)$, yielding $t = 3$ and $t = 2$. Therefore, $\cos x = 6^{-\sqrt{3}}$ and $6^{-\sqrt{2}}$.
Correct Answer: 8