Question

The value of $ \tan^{-1} \left( \frac{x}{y} \right) - \tan^{-1} \left( \frac{x - y}{x + y} \right) \text{ is} $

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. None of these

Hint:

When dealing with the difference of inverse tangents, use the standard identity for simplifying the expression.

Answer 1

The Correct Option is A

We are given the expression: \[ \tan^{-1} \left( \frac{x}{y} \right) - \tan^{-1} \left( \frac{x - y}{x + y} \right) \]
Step 1: Use the formula for the difference of arctangents
We use the standard identity for the difference of two inverse tangents: \[ \tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a - b}{1 + ab} \right) \] Let \( a = \frac{x}{y} \) and \( b = \frac{x - y}{x + y} \).
Applying the identity: \[ \tan^{-1} \left( \frac{x}{y} \right) - \tan^{-1} \left( \frac{x - y}{x + y} \right) = \tan^{-1} \left( \frac{\frac{x}{y} - \frac{x - y}{x + y}}{1 + \frac{x}{y} \cdot \frac{x - y}{x + y}} \right) \]
Step 2: Simplify the expression
After simplification, the result simplifies to \( \frac{\pi}{2} \), which is the required value.
Step 3: Conclusion
Thus, the correct answer is \( \frac{\pi}{2} \), which corresponds to option (a).