Question

If \( \tan^{-1}x + \tan^{-1}y = \frac{\pi}{4} \), then what is the value of \(x + y + xy\)?

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-1\)

Hint:

Since the equation must hold for many different pairs of x and y, but the expression x + y + xy always works out to the same number, try substituting one simple valid pair first, such as y = 0, to find the value quickly, then check a second pair for confidence.

Answer 1

The Correct Option is B

Topic - Inverse Trigonometric Functions:
This problem uses the addition property of inverse tangent functions to find a relationship between variables.
Step 1: Understanding the Question:
Given the sum of two inverse tangents equals \(\pi/4\), we must calculate the specific algebraic expression \(x + y + xy\).
Step 2: Key Formula or Approach:
The identity for addition is: \[ \tan^{-1}x + \tan^{-1}y = \tan^{-1} \left( \frac{x+y}{1-xy} \right) \] Step 3: Detailed Solution:
1. Apply the identity:
\[ \tan^{-1} \left( \frac{x+y}{1-xy} \right) = \frac{\pi}{4} \] 2. Take the tangent of both sides:
\[ \frac{x+y}{1-xy} = \tan \left( \frac{\pi}{4} \right) \] 3. Since \(\tan(\pi/4) = 1\):
\[ \frac{x+y}{1-xy} = 1 \] 4. Rearrange the equation:
\[ x + y = 1 - xy \] \[ x + y + xy = 1 \] Step 4: Final Answer:
The value is 1.