Question

Find the value of the composite inverse trigonometric expression: \( \cot^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2}\right)\right] \)

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

Hint:

Always work inward-out for inverse trigonometric compositions. Treat each layer as finding a simple angle, evaluating its standard value before moving to the next operator.

Answer 1

The Correct Option is B

Concept: To solve composite inverse trigonometric functions, evaluate the expression step-by-step from the innermost parenthesis outward using standard principal values.

Step 1: Evaluate the innermost inverse sine term. We know that the principal value branch of \( \sin^{-1}x \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Since \( \sin(\frac{\pi}{6}) = \frac{1}{2} \): \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

Step 2: Multiply by the coefficient and evaluate the cosine function. Substitute this value into the next layer of the expression: \[ 2 \cdot \left(\sin^{-1}\frac{1}{2}\right) = 2 \cdot \frac{\pi}{6} = \frac{\pi}{3} \] Now evaluate the outer cosine function at this angle: \[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Step 3: Evaluate the outermost cotangent inverse function. Multiply the result by the remaining coefficient scalar \( 2 \): \[ 2 \cdot \cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1 \] Finally, compute the principal value of the inverse cotangent function: \[ \cot^{-1}(1) = \frac{\pi}{4} \] Let's check the cotangent values to ensure accuracy. Since \( \cot(\frac{\pi}{4}) = 1 \), the final angle simplifies perfectly to \( \frac{\pi}{4} \). .