Question

The value of \( \cot^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2}\right)\right] \) is:

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{2\pi}{3} \)
• D. \( \frac{\pi}{3} \)

Hint:

Whenever expressions involve inverse trigonometric functions:
• First convert the inverse trigonometric quantity into a standard angle.
• Then simplify step-by-step using standard trigonometric values.
• Remember principal values: \[ \sin^{-1}x \in \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \] \[ \cot^{-1}x \in (0,\pi) \] Important standard values: \[ \sin^{-1}\frac12=\frac{\pi}{6} \] \[ \cos\frac{\pi}{3}=\frac12 \] \[ \cot^{-1}(1)=\frac{\pi}{4} \]

Answer 1

The Correct Option is D

Concept: This problem is based on inverse trigonometric functions together with standard trigonometric identities. The important identities used are: \[ \cos 2\theta = 1-2\sin^2\theta \] and \[ \cot^{-1}x=\theta \iff \cot\theta=x \] where the principal value of \( \cot^{-1}x \) lies in: \[ (0,\pi) \] The strategy is:
• First simplify the inverse trigonometric expression.
• Then evaluate the trigonometric function inside the bracket.
• Finally determine the angle whose cotangent gives the obtained value.

Step 1: Evaluating the inverse sine expression We are given: \[ \sin^{-1}\frac12 \] We know that: \[ \sin\frac{\pi}{6}=\frac12 \] Therefore, \[ \sin^{-1}\frac12=\frac{\pi}{6} \] Substituting this into the given expression: \[ \cot^{-1}\left[2\cos\left(2\times \frac{\pi}{6}\right)\right] \] \[ =\cot^{-1}\left[2\cos\frac{\pi}{3}\right] \]

Step 2: Evaluating the cosine value We know: \[ \cos\frac{\pi}{3}=\frac12 \] Therefore, \[ 2\cos\frac{\pi}{3} = 2\times \frac12 = 1 \] Thus the expression becomes: \[ \cot^{-1}(1) \]

Step 3: Finding the angle whose cotangent is 1 We now determine the principal value angle \(\theta\) such that: \[ \cot\theta=1 \] We know: \[ \cot\frac{\pi}{4}=1 \] Therefore, \[ \cot^{-1}(1)=\frac{\pi}{4} \]

Step 4: Comparing with the options Hence the value of the given expression is: \[ \boxed{\frac{\pi}{4}} \] Therefore, the correct option is: \[ \boxed{(B)} \]