Question

Evaluate: \[ \tan (\cos^{-1} \frac{4}{5}) + \tan^{-1} \frac{2}{3} \]

• A. \( \frac{6}{17} \)
• B. \( \frac{7}{16} \)
• C. \( \frac{16}{7} \)
• D. None of these

Hint:

Use the identity \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

Answer 1

The Correct Option is D

Step 1: Finding \( \tan (\cos^{-1} \frac{4}{5}) \).
Using the identity: \[ \tan (\cos^{-1} x) = \frac{\sqrt{1 - x^2}}{x} \] Substituting \( x = \frac{4}{5} \): \[ \tan (\cos^{-1} \frac{4}{5}) = \frac{\sqrt{1 - \left(\frac{4}{5}\right)^2}}{\frac{4}{5}} \] \[ = \frac{\sqrt{\frac{25}{25} - \frac{16}{25}}}{\frac{4}{5}} \] \[ = \frac{\sqrt{9/25}}{4/5} = \frac{3/5}{4/5} = \frac{3}{4} \] Step 2: Computing total expression.
Using sum of tangent formula: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Substituting values \( A = \cos^{-1} \frac{4}{5} \), \( B = \tan^{-1} \frac{2}{3} \): \[ \tan A = \frac{3}{4}, \quad \tan B = \frac{2}{3} \] \[ \tan(A + B) = \frac{\frac{3}{4} + \frac{2}{3}}{1 - \left(\frac{3}{4} \times \frac{2}{3}\right)} \] \[ = \frac{\frac{9}{12} + \frac{8}{12}}{1 - \frac{6}{12}} \] \[ = \frac{17}{6} \neq \frac{6}{17}, \frac{7}{16}, \frac{16}{7} \] Thus, the correct answer is (D).