Question

Evaluate: \[ \tan^{-1} 2 + \tan^{-1} 3 \]

• A. \( \frac{-\pi}{4} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{3\pi}{4} \)
• D. \( \frac{5\pi}{4} \)

Hint:

Use the formula \( \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \) for summing inverse tangents.

Answer 1

The Correct Option is C

Using the identity: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right), \quad \text{if } ab1 \] Substituting \( a = 2, b = 3 \): \[ \tan^{-1} 2 + \tan^{-1} 3 = \tan^{-1} \left( \frac{2 + 3}{1 - (2 \times 3)} \right) \] \[ = \tan^{-1} \left( \frac{5}{1 - 6} \right) = \tan^{-1} \left( \frac{5}{-5} \right) = \tan^{-1} (-1) \] \[ = -\frac{\pi}{4} \] Since the angle must be in the second quadrant: \[ \tan^{-1} 2 + \tan^{-1} 3 = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \] Thus, the correct answer is \( \frac{3\pi}{4} \).