Question

The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :

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

Hint:

Always be mindful of the principal value ranges for inverse trigonometric functions. A common mistake is to choose an angle in the wrong quadrant. For \( \sec^{-1}(x) \), it's often easier to convert the problem to \( \cos^{-1}(1/x) \) and use its well-known range.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to evaluate the given expression by finding the principal values of the two inverse trigonometric functions. The principal value is the unique value of the angle within the defined range of the inverse trigonometric function.
Step 2: Key Formula or Approach:
The principal value ranges are:
- For \( \tan^{-1}(x) \), the range is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).
- For \( \sec^{-1}(x) \), the range is \( [0, \pi] - \{\frac{\pi}{2}\} \).
Step 3: Detailed Explanation or Calculation:
Part 1: Evaluate \( \tan^{-1}(\sqrt{3}) \)
Let \( \theta = \tan^{-1}(\sqrt{3}) \). This means \( \tan(\theta) = \sqrt{3} \), where \( \theta \in (-\frac{\pi}{2}, \frac{\pi}{2}) \).
The angle in this range for which the tangent is \( \sqrt{3} \) is \( \frac{\pi}{3} \).
So, \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
Part 2: Evaluate \( \sec^{-1}(-2) \)
Let \( \phi = \sec^{-1}(-2) \). This means \( \sec(\phi) = -2 \), where \( \phi \in [0, \pi] \) and \( \phi \neq \frac{\pi}{2} \).
\( \sec(\phi) = -2 \) is equivalent to \( \cos(\phi) = -\frac{1}{2} \).
We need to find the angle \( \phi \) in the range \( [0, \pi] \) where the cosine is \( -\frac{1}{2} \). This occurs in the second quadrant.
The reference angle is \( \cos^{-1}(\frac{1}{2}) = \frac{\pi}{3} \). For the second quadrant, the angle is \( \pi - \frac{\pi}{3} = \frac{2\pi}{3} \).
So, \( \sec^{-1}(-2) = \frac{2\pi}{3} \).
Part 3: Combine the results
\[ \tan^{-1}\sqrt{3} - \sec^{-1}(-2) = \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3} \] Step 4: Final Answer:
The value of the expression is \( -\frac{\pi}{3} \).