Question

Evaluate \( \sin \left( \tan^{-1}\frac{4}{5} + \tan^{-1}\frac{4}{3} + \tan^{-1}\frac{1}{9} - \tan^{-1}\frac{1}{7} \right) \):

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( 1 \)

Hint:

For trigonometric sums involving inverse functions, use addition and subtraction formulas to simplify the expressions before evaluating.