Question

If \( \cos \cot^{-1} \left( \frac{1}{2} \right) = \cot (\cos^{-1} x) \), then the value of \( x \) is:

• A. \( \frac{1}{\sqrt{6}} \)
• B. \( \frac{-1}{\sqrt{12}} \)
• C. \( \frac{2}{\sqrt{6}} \)
• D. \( \frac{-2}{\sqrt{6}} \)

Hint:

To convert inverse trigonometric expressions, use the basic definitions of trigonometric functions in right-angled triangles.

Answer 1

The Correct Option is A

Step 1: {Express \( \cot^{-1} \) in terms of cosine}
Let \[\alpha = \cot^{-1} \left( \frac{1}{2} \right).\]Then, \[\cot \alpha = \frac{1}{2} \Rightarrow \cos \alpha = \frac{1}{\sqrt{5}}.\]Step 2: {Use cotangent identity}
We use the identity \[\cos (\cos^{-1} x) = \cot \left( \cos^{-1} x \right).\]This implies \[\cot (\cos^{-1} x) = \frac{x}{\sqrt{1 - x^2}}.\]Step 3: {Equating both sides}
We equate the results from Step 1 and Step 2:\[\frac{1}{\sqrt{5}} = \frac{x}{\sqrt{1 - x^2}}.\]Squaring both sides yields:\[1 - x^2 = 5x^2.\]Step 4: {Solve for \( x \)}
Rearranging the equation:\[1 = 6x^2.\]The solutions for \( x \) are:\[x = \pm \frac{1}{\sqrt{6}}.\]Step 5: {Select the correct sign}
We disregard the negative solution:\[x = \frac{1}{\sqrt{6}}.\]