Question

If $\sec^{-1}\left(\frac{x}{x+2}\right) = \frac{\pi}{2} - \csc^{-1}\left(\frac{1}{2}\right)$, then $x = $ ________.

• A. -2
• B. -4
• C. 2
• D. 4
• E. -1

Hint:

Always check the domain: $\sec^{-1}(y)$ requires $|y| \ge 1$. Here, $|-4/(-4+2)| = 2$, which is valid.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The question presents an equation involving inverse trigonometric functions. To solve it, we must analyze the domains and ranges of the functions involved and use standard identities if applicable.
Step 2: Key Formula or Approach
The key identity for inverse secant and cosecant is \(\sec^{-1}(y) + \csc^{-1}(y) = \frac{\pi}{2}\), which is valid for \(|y| \geq 1\).
Also, the domain of \(\csc^{-1}(y)\) is \(|y| \geq 1\), meaning \(y \geq 1\) or \(y \leq -1\).
Step 3: Detailed Explanation
1. Analyze the given equation.
The equation is \(\sec^{-1}\left(\frac{x}{x+2}\right) = \frac{\pi}{2} - \csc^{-1}\left(\frac{1}{2}\right)\).
Let's focus on the term \(\csc^{-1}\left(\frac{1}{2}\right)\).
2. Check the domain of the inverse cosecant function.
The domain of the function \(\csc^{-1}(y)\) is the set of all real numbers \(y\) such that \(|y| \geq 1\). This means \(y\) must be in the interval \((-\infty, -1] \cup [1, \infty)\).
In our problem, we have \(y = \frac{1}{2} = 0.5\).
Since \(|0.5|<1\), the value \(0.5\) is not in the domain of the \(\csc^{-1}\) function.
3. Conclusion.
Because the term \(\csc^{-1}\left(\frac{1}{2}\right)\) is undefined, the entire equation is invalid as written. There is no real angle \(\theta\) such that \(\csc(\theta) = \frac{1}{2}\) (since the range of \(\csc(\theta)\) is \((-\infty, -1] \cup [1, \infty)\)).
Therefore, the question has no solution and is flawed. This is why it was likely cancelled in the examination.
Step 4: Final Answer
The question is invalid because the expression \(\csc^{-1}\left(\frac{1}{2}\right)\) is undefined.