Question

Evaluate: \(\sin \left( \cot^{-1} x \right)\)

• A. \(\sqrt{1 + x^2}\)
• B. \(x\)
• C. \((1 + x^2)^{-\frac{3}{2}}\)
• D. \(\frac{1}{\sqrt{1 + x^2}}\)

Hint:

For inverse cotangent functions, remember the identity \(\sin(\cot^{-1} x) = \frac{1}{\sqrt{1 + x^2}}\).

Answer 1

The Correct Option is D

Step 1: Let \(\theta = \cot^{-1} x\). This means \(\cot \theta = x\). Step 2: Use the identity \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), which leads to: \[ \cot^2 \theta + \sin^2 \theta = 1 \quad \Rightarrow \quad \sin^2 \theta = \frac{1}{1 + x^2} \] Thus, \[ \sin \theta = \frac{1}{\sqrt{1 + x^2}} \] Step 3: Therefore, \[ \sin \left( \cot^{-1} x \right) = \frac{1}{\sqrt{1 + x^2}} \] ---