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. \( \text{cosec}^{-1} : \mathbb{R} - (-1, 1) \to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)

Hint:

Always remember the "holes" in reciprocal trig functions: sec excludes \( \pi/2 \), cosec excludes \( 0 \).
Think of the graphs: wherever the original trig function has a vertical asymptote, that point is excluded from the range of the inverse.

Answer 1

The Correct Option is D

Concept:
• Principal value branches are the specific ranges defined to make trigonometric functions bijective so their inverses can exist.
• We must identify which range correctly excludes points where the original function is undefined (division by zero).

Step 1: Evaluate the standard branches for options (A), (B), and (C)

• \( \tan^{-1} x \): The range is the open interval \( (-\pi/2, \pi/2) \). Correct.
• \( \sec^{-1} x \): Since \( \sec \theta = 1/\cos \theta \), the value \( \pi/2 \) (where \( \cos \theta = 0 \)) must be excluded from \( [0, \pi] \). Correct.
• \( \cot^{-1} x \): The range is the open interval \( (0, \pi) \). Correct.

Step 2: Analyze the definition of \( \text{cosec}^{-1} x \) in option (D)
The function \( \text{cosec } \theta = 1/\sin \theta \). The function is undefined when \( \sin \theta = 0 \), which occurs at \( \theta = 0 \) within the interval \( [-\pi/2, \pi/2] \). Therefore, the principal value branch must be \( [-\pi/2, \pi/2] - \{0\} \).

Step 3: Identify the error
Option (D) provides the range as \( [-\pi/2, \pi/2] \) but fails to exclude the singular point \( \{0\} \). Thus, it is not a correctly defined principal value branch.