Question

The value of $\tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3$ is:}

• A. $3\pi/4$
• B. $\pi/2$
• C. $\pi$
• D. $2\pi$

Hint:

Standard Result: $\tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3 = \pi$.

Answer 1

The Correct Option is C

Step 1: Concept
Use the addition formula for inverse tangents: $\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}(\frac{x+y}{1-xy})$ if $xy > 1$.

Step 2: Meaning
Calculate $\tan^{-1} 2 + \tan^{-1} 3$ first. Since $2 \times 3 = 6 > 1$, we add $\pi$.

Step 3: Analysis
$\tan^{-1} 2 + \tan^{-1} 3 = \pi + \tan^{-1}(\frac{2+3}{1-6}) = \pi + \tan^{-1}(-1) = \pi - \pi/4 = 3\pi/4$.

Step 4: Conclusion
Adding $\tan^{-1} 1$ (which is $\pi/4$): $\pi/4 + 3\pi/4 = \pi$.
Final Answer: (C)