`tan^(-1)x-tan^(-1)y=tan^(-1)((x-y)/(1+xy))` holds good for

Options

A. All `x, y in R`
B. `absx lt 1absy lt 1`
C. `absx gt1,absygt1`
D. `xy gt -1`

Answer

To determine when the equation \( \tan^{-1} x - \tan^{-1} y = \tan^{-1} \left( \frac{x - y}{1 + xy} \right) \) holds true, we can follow these steps:

### Step 1: Understand the Formula
The formula for the difference of inverse tangent functions is given by:
\[ \tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a - b}{1 + ab} \right) \]
This formula is valid under certain conditions.

### Step 2: Identify the Conditions
For the formula to hold, the denominator \(1 + ab\) must be greater than zero:
\[ 1 + xy > 0 \]
This implies:
\[ xy > -1 \]

### Step 3: Conclusion
Thus, the condition under which the equation \( \tan^{-1} x - \tan^{-1} y = \tan^{-1} \left( \frac{x - y}{1 + xy} \right) \) holds true is:
\[ xy > -1 \]

### Final Answer
The equation holds true when \( xy > -1 \).

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