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
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
Show Hint
When summing multiple arctangents, apply the addition formula step-by-step to simplify the expression.
Updated On: Mar 28, 2026
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{12} \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{6} \)
Show Solution
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The Correct Option is C
Solution and Explanation
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).
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).
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