Problem
Solve: \( \cos^{-1}(\sqrt{3}x) + \cos^{-1}(x) = \frac{\pi}{2} \)
Solution
Step 1: Use identity
If: \[ \cos^{-1}(a) + \cos^{-1}(b) = \frac{\pi}{2} \] then: \[ a^2 + b^2 = 1 \]
Step 2: Apply
\[ (\sqrt{3}x)^2 + x^2 = 1 \]
\[ 3x^2 + x^2 = 1 \]
\[ 4x^2 = 1 \Rightarrow x^2 = \frac{1}{4} \]
\[ x = \pm \frac{1}{2} \]
Step 3: Domain check
For \( \cos^{-1}(\sqrt{3}x) \), we need: \[ -1 \le \sqrt{3}x \le 1 \Rightarrow |x| \le \frac{1}{\sqrt{3}} \]
Check:
- \( x = \frac{1}{2} \) ✔ valid
- \( x = -\frac{1}{2} \) ✔ valid
Final Answer
\[ \boxed{x = \pm \frac{1}{2}} \]
Explanation
Using identity cos⁻¹a + cos⁻¹b = π/2 ⇒ a² + b² = 1 simplifies the equation directly.