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: Concept
Recall the identity $\sec^{-1}(y) + \csc^{-1}(y) = \frac{\pi}{2}$ for $|y| \ge 1$. This implies $\frac{\pi}{2} - \csc^{-1}(y) = \sec^{-1}(y)$.

Step 2: Interpretation
The term $\csc^{-1}(1/2)$ is mathematically undefined for real numbers. However, based on the options, it is likely a typo for $\csc^{-1}(2)$ or $\sin^{-1}(1/2)$. Given $\frac{\pi}{2} - \sin^{-1}(1/2) = \frac{\pi}{3}$, the equation becomes $\sec^{-1}(\frac{x}{x+2}) = \frac{\pi}{3}$.

Step 3: Solving for x
$\frac{x}{x+2} = \sec\left(\frac{\pi}{3}\right) = 2$
$x = 2(x + 2) \implies x = 2x + 4$
$-x = 4 \implies x = -4$.

Step 4: Conclusion
Substituting $x = -4$ into the original expression gives an argument of $2$, which is within the valid domain for $\sec^{-1}$. Final Answer: (B)