Question

The value of $tan^{-1} (\frac{7}4) - tan^{-1} (\frac{3}{11})$ is equal to 

• A. $\frac{-π}3$
• B. $\frac{-π}4$
• C. $\frac{π}4$
• D. $\frac{π}3$
• E. $π$

Answer 1

The Correct Option is C

Given expression: 

\(\tan^{-1} \left(\frac{7}{4} \right) - \tan^{-1} \left(\frac{3}{11} \right)\)

Using the identity:

\(\tan^{-1} A - \tan^{-1} B = \tan^{-1} \left(\frac{A - B}{1 + AB} \right)\), when \(AB < 1\)

Here, \(A = \frac{7}{4}\) and \(B = \frac{3}{11}\).

Calculating \(A - B\):

\(\frac{7}{4} - \frac{3}{11} = \frac{(7 \times 11) - (3 \times 4)}{4 \times 11}\)

\(= \frac{77 - 12}{44} = \frac{65}{44}\)

Calculating \(1 + AB\):

\(1 + \left(\frac{7}{4} \times \frac{3}{11}\right)\)

\(= 1 + \frac{21}{44}\)

\(= \frac{44}{44} + \frac{21}{44} = \frac{65}{44}\)

Using the identity:

\(\tan^{-1} \left(\frac{65}{44} \div \frac{65}{44} \right) = \tan^{-1} (1)\)

\(\tan^{-1} (1) = \frac{\pi}{4}\)

Thus, the correct answer is:

\(\frac{\pi}{4}\)