<p>If \(\sin(\cot^{-1}(x+1)) = \cos(\tan^{-1} x)\), then the value of x is</p>
Step-by-Step Solution
Key Concept: Convert inverse trigonometric expressions to right triangle representations and equate the resulting expressions.
<p><strong>Step 1:</strong> Let \(\cot^{-1}(x+1) = \alpha\), so \(\cot \alpha = x+1\)</p><p><strong>Step 2:</strong> In a right triangle with \(\cot \alpha = x+1\), we have \(\sin \alpha = \frac{1}{\sqrt{(x+1)^2+1}}\)</p><p><strong>Step 3:</strong> Let \(\tan^{-1} x = \beta\), so \(\tan \beta = x\)</p><p><strong>Step 4:</strong> Then \(\cos \beta = \frac{1}{\sqrt{x^2+1}}\)</p><p><strong>Step 5:</strong> From the given equation: \(\frac{1}{\sqrt{(x+1)^2+1}} = \frac{1}{\sqrt{x^2+1}}\)</p><p><strong>Step 6:</strong> This gives \((x+1)^2 + 1 = x^2 + 1\), which simplifies to \(x^2 + 2x + 1 = x^2\)</p><p><strong>Step 7:</strong> Therefore \(2x + 1 = 0\), so \(x = -\frac{1}{2}\)</p>
Correct Answer: A