<p>Solve the equation \(\cos^{-1}\sqrt{x^2 - 25} = \cos^{-1}\frac{\sqrt{x^2-25}}{x}\) for \(x > 12\)</p>
Step-by-Step Solution
Key Concept: Use the condition that for inverse cosine functions to be equal, their arguments must be equal, and apply domain restrictions to find the unique solution.
<p><strong>Step 1:</strong> From the given equation, we need $\sqrt{x^2 - 25} = \frac{\sqrt{x^2-25}}{x}$</p><p><strong>Step 2:</strong> For $x > 12$ and $\sqrt{x^2 - 25} > 0$, dividing both sides by $\sqrt{x^2-25}$:</p><p>$$1 = \frac{1}{x}$$</p><p>This would give $x = 1$, which contradicts $x > 12$.</p><p><strong>Step 3:</strong> Alternatively, squaring both sides: $x^2 - 25 = 12^2$</p><p>$$x^2 = 169$$</p><p>$$x = \pm 13$$</p><p><strong>Step 4:</strong> Since $x > 12$, we have $x = 13$</p><p>∴ The answer is <strong>13</strong>.</p>
Correct Answer: 13