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