Question

$\tan^{-1}(\frac{1001}{999}) - \tan^{-1}(\frac{2}{2000}) = $ ________.

• A. $\frac{\pi}{3}$
• B. $\pi$
• C. 1
• D. $\frac{\pi}{6}$
• E. $\frac{\pi}{4}$

Hint:

Simplify the argument before calculating the inverse tangent.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
This problem involves simplifying an expression with inverse tangent functions. We will use the formula for the difference of two inverse tangents.
Step 2: Key Formula or Approach
The key formula is:
\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x-y}{1+xy}\right) \] This formula is valid when \(xy>-1\).
Step 3: Detailed Explanation
1. Simplify the second term.
The expression is \(\tan^{-1}\left(\frac{1001}{999}\right) - \tan^{-1}\left(\frac{2}{2000}\right)\).
The second term can be simplified:
\[ \tan^{-1}\left(\frac{2}{2000}\right) = \tan^{-1}\left(\frac{1}{1000}\right) \] 2. Apply the difference formula.
Let \(x = \frac{1001}{999}\) and \(y = \frac{1}{1000}\). Both x and y are positive, so \(xy>-1\). We can apply the formula.
The expression becomes:
\[ \tan^{-1}\left(\frac{\frac{1001}{999} - \frac{1}{1000}}{1 + \left(\frac{1001}{999}\right)\left(\frac{1}{1000}\right)}\right) \] 3. Simplify the argument of the inverse tangent.
Let's compute the numerator and denominator separately.
- Numerator: \[ \frac{1001}{999} - \frac{1}{1000} = \frac{(1001)(1000) - (999)(1)}{999 \times 1000} = \frac{1001000 - 999}{999000} = \frac{1000001}{999000} \] - Denominator: \[ 1 + \frac{1001}{999000} = \frac{999000}{999000} + \frac{1001}{999000} = \frac{999000 + 1001}{999000} = \frac{1000001}{999000} \] Now, divide the numerator by the denominator:
\[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{1000001}{999000}}{\frac{1000001}{999000}} = 1 \] 4. Evaluate the final expression.
The entire expression simplifies to:
\[ \tan^{-1}(1) \] The principal value of \(\tan^{-1}(1)\) is the angle \(\theta\) in \((-\pi/2, \pi/2)\) such that \(\tan(\theta) = 1\). This angle is \(\frac{\pi}{4}\).
Step 4: Final Answer
The value of the expression is \(\frac{\pi}{4}\).