Question

The value of \(3tan^{-1}(\frac{1}{2}) = \ldots\)

• A. \(tan^{-1}(\frac{5}{2})\)
• B. \(tan^{-1}(\frac{2}{5})\)
• C. \(cot^{-1}(\frac{11}{2})\)
• D. \(tan^{-1}(\frac{11}{2})\)

Hint:

Use the double angle formula for \(\tan^{-1}\), then add one more \(\tan^{-1}\frac12\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Use \(2\tan^{-1}x=\tan^{-1}\dfrac{2x}{1-x^2}\) for \(|x|<1\), and \(\tan^{-1}a+\tan^{-1}b=\tan^{-1}\dfrac{a+b}{1-ab}\) for \(ab<1\).

Step 2: Key Formula or Approach
\[ 2\tan^{-1}\frac12=\tan^{-1}\frac{1}{1-\frac14}=\tan^{-1}\frac43 \]

Step 3: Detailed Explanation
\[ 3\tan^{-1}\frac12=\tan^{-1}\frac43+\tan^{-1}\frac12=\tan^{-1}\frac{\frac43+\frac12}{1-\frac23} \]
\[ =\tan^{-1}\frac{11/6}{1/3}=\tan^{-1}\frac{11}{2} \]
The product \(\tfrac43\cdot\tfrac12=\tfrac23<1\), so the formula applies directly.

Final Answer:
The value is \(\tan^{-1}\frac{11}{2}\), option (D). \[ \boxed{\tan^{-1}\dfrac{11}{2}\ \text{(D)}} \]