Question

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

• A. \( \tan^{-1} : \mathbb{R} \to \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
• B. \( \sec^{-1} : \mathbb{R} - (-1, 1) \to [0, \pi] - \left\{ \frac{\pi}{2} \right\} \)
• C. \( \cot^{-1} : \mathbb{R} \to (0, \pi) \)
• D. \( \csc^{-1} : \mathbb{R} - (-1, 1) \to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)

Hint:

Memory trick for ranges:
- Group 1 (\( \sin^{-1}, \tan^{-1}, \csc^{-1} \)) are related to \( [-\pi/2, \pi/2] \).
- Group 2 (\( \cos^{-1}, \cot^{-1}, \sec^{-1} \)) are related to \( [0, \pi] \).
Remember to exclude points where the original function is undefined (denominator zero).

Answer 1

The Correct Option is D

Concept:
• Principal Value Branches refer to the range of the inverse trigonometric functions within which the function is one-to-one and onto.
• These ranges are standardized to maintain consistency in calculations.

Step 1: Verify the branch for \( \tan^{-1} \) and \( \cot^{-1} \)
The range of \( \tan^{-1} x \) is the open interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \).
Thus, (A) is correctly defined.
The range of \( \cot^{-1} x \) is the open interval \( (0, \pi) \).
Thus, (C) is correctly defined.

Step 2: Verify the branch for \( \sec^{-1} \)
The domain of \( \sec^{-1} x \) is \( (-\infty, -1] \cup [1, \infty) \), often written as \( \mathbb{R} - (-1, 1) \).
Its range is \( [0, \pi] \) excluding the value where \( \cos \theta = 0 \), which is \( \frac{\pi}{2} \).
Range of \( \sec^{-1} = [0, \pi] - \left\{ \frac{\pi}{2} \right\} \).
Thus, (B) is correctly defined.

Step 3: Verify the branch for \( \csc^{-1} \)
The domain of \( \csc^{-1} x \) is \( \mathbb{R} - (-1, 1) \).
Its range is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \) excluding the value where \( \sin \theta = 0 \), which is \( 0 \).
Correct Range of \( \csc^{-1} = \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] - \{0\} \).
Option (D) provides the range as \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \) without excluding \( \{0\} \).
Therefore, (D) is not correctly defined.