Question

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

• A. \(\frac{3\pi}{4}\)
• B. \(\frac{\pi}{2}\)
• C. \(\pi\)
• D. \(2\pi\)

Hint:

While adding inverse tangent terms, always check the sign of \(1-ab\) to decide the correct quadrant.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to sum three inverse tangent values. This is a common property problem in inverse trigonometry.
Step 2: Key Formula or Approach:
The addition formula for \( \tan^{-1} x + \tan^{-1} y \):
1. If \( xy<1 \): \( \tan^{-1} x + \tan^{-1} y = \tan^{-1}(\frac{x+y}{1-xy}) \).
2. If \( xy>1 \) and \( x, y>0 \): \( \tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}(\frac{x+y}{1-xy}) \).
Step 3: Detailed Explanation:

We know \( \tan^{-1} 1 = \pi/4 \).

Now consider \( \tan^{-1} 2 + \tan^{-1} 3 \).

Here \( x = 2 \) and \( y = 3 \). Product \( xy = 6 \), which is \(>1 \).

Apply the second case of the addition formula:
\[ \tan^{-1} 2 + \tan^{-1} 3 = \pi + \tan^{-1}\left(\frac{2+3}{1-2(3)}\right) \]
\[ = \pi + \tan^{-1}\left(\frac{5}{-5}\right) = \pi + \tan^{-1}(-1) \]

Since \( \tan^{-1}(-1) = -\pi/4 \):
\[ \tan^{-1} 2 + \tan^{-1} 3 = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

Finally, sum all three terms:
\[ \tan^{-1} 1 + (\tan^{-1} 2 + \tan^{-1} 3) = \frac{\pi}{4} + \frac{3\pi}{4} = \frac{4\pi}{4} = \pi \]

Step 4: Final Answer:
The sum is \( \pi \).