Question

The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{5} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{8} \right) \] is

• A. \( \frac{\pi}{3} \)
• B. \( \frac{\pi}{12} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

When summing multiple arctangents, apply the addition formula step-by-step to simplify the expression.

Answer 1

The Correct Option is C

Step 1: Use the addition formula for arctangent.
We can use the identity for the sum of two arctangents: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \] We apply this identity sequentially to sum all four terms in the given expression. After performing the calculations, we find that the sum of these arctangents is \( \frac{\pi}{4} \).

Step 2: Conclusion.
Thus, the correct answer is \( \frac{\pi}{4} \), corresponding to option (C).