Question

The value of \( \sin(\cot^{-1}(x)) \) is:

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

To evaluate trigonometric functions of inverse trigonometric expressions, construct a right triangle using the given ratios and apply fundamental definitions.

Answer 1

The Correct Option is A

To find \( \sin(\cot^{-1}(x)) \), we follow these steps: Step 1: Define the angle Let \( \theta = \cot^{-1}(x) \). By the definition of the inverse cotangent function, we have: \[ \cot(\theta) = x \] This implies that: \[ \theta \in (0, \pi) \quad \text{(principal range of the cotangent inverse function)}. \]
Step 2: Represent in terms of a right triangle From the definition of cotangent: \[ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \] Assign values based on the given \( \cot(\theta) = x \): \[ \text{adjacent side} = x, \quad \text{opposite side} = 1. \]
Step 3: Calculate the hypotenuse Using the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{\text{adjacent}^2 + \text{opposite}^2} = \sqrt{x^2 + 1}. \]
Step 4: Find \( \sin(\theta) \) From the definition of sine: \[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}. \] Substituting the values: \[ \sin(\theta) = \frac{1}{\sqrt{x^2 + 1}}. \]
Step 5: Final result Since \( \theta = \cot^{-1}(x) \), we conclude: \[ \sin(\cot^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}. \]
Final Answer: \[ \boxed{\frac{1}{\sqrt{1 + x^2}}}. \]