Question

Evaluate: \[ \tan^{-1}\left(\frac12\right) +\tan^{-1}\left(\frac13\right) +\tan^{-1}\left(\frac23\right) +\tan^{-1}\left(\frac15\right) \]

• A. \(\dfrac{\pi}{4}\)
• B. \(\tan^{-1}\left(\dfrac{7}{11}\right)\)
• C. \(\dfrac{\pi}{2}\)
• D. \(\tan^{-1}\left(\dfrac{23}{24}\right)\)

Hint:

Look for pairs satisfying \[ \frac{a+b}{1-ab}=1. \] Such pairs immediately simplify to \(\tan^{-1}(1)=\frac{\pi}{4}\).

Answer 1

The Correct Option is C