Question

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

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

Hint:

Always look out for points where the original base functions are undefined. For example, \( \text{cosec}(\theta) \) is undefined at \( \theta = 0 \), so \( 0 \) must always be excluded from the principal range of \( \text{cosec}^{-1}x \).

Answer 1

The Correct Option is D

Concept: The principal value branches for inverse trigonometric functions define specific restricted codomains to make the original trigonometric functions bijective. Let's recall the standard, universally established principal value branches:
• \( \tan^{-1}x \): Domain is \( \mathbb{R} \), Range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
• \( \sec^{-1}x \): Domain is \( \mathbb{R} - (-1, 1) \), Range is \( [0, \pi] - \left\{\frac{\pi}{2}\right\} \)
• \( \cot^{-1}x \): Domain is \( \mathbb{R} \), Range is \( (0, \pi) \)
• \( \text{cosec}^{-1}x \): Domain is \( \mathbb{R} - (-1, 1) \), Range must exclude \( 0 \) because \( \text{cosec}(0) \) is undefined. Thus, its true range is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\} \).

Step 1: Check option (A). The given branch is \( \tan^{-1} : \mathbb{R} \rightarrow \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \). This matches the standard definition precisely, so it is correct.

Step 2: Check option (B). The given branch is \( \sec^{-1} : \mathbb{R} - (-1, 1) \rightarrow [0, \pi] - \left\{ \frac{\pi}{2} \right\} \). This matches the mathematical standard definitions exactly because \( \cos(\pi/2) = 0 \), hence secant is undefined there. This is correct.

Step 3: Check option (C). The given branch is \( \cot^{-1} : \mathbb{R} \rightarrow (0, \pi) \). This is correct according to modern standard conventions.

Step 4: Check option (D). The given branch is listed as \( \text{cosec}^{-1} : \mathbb{R} - (-1, 1) \rightarrow \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). However, the value \( 0 \) lies within the interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). Since \( \sin(0) = 0 \), \( \text{cosec}(0) = \frac{1}{0} \) is totally undefined. Therefore, \( 0 \) cannot belong to the codomain of the principal branch of \( \text{cosec}^{-1} \). The true branch must be written as \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\} \). Consequently, option (D) is not correctly defined.