If $ an^{-1}\left(\dfrac{1}{2} ight) + an^{-1}\left(\dfrac{1}{3} …

Question

If $	an^{-1}\left(\dfrac{1}{2}ight) + 	an^{-1}\left(\dfrac{1}{3}ight) = ?$

Accepted Answer

D. $\dfrac{\pi}{4}$

Use the identity $	an^{-1}(a) + 	an^{-1}(b) = 	an^{-1}\left(\dfrac{a + b}{1 - ab}ight)$, valid when $ab < 1$.
With $a = 1/2$, $b = 1/3$: $ab = 1/6 < 1$, so the identity applies cleanly.
$\dfrac{a + b}{1 - ab} = \dfrac{1/2 + 1/3}{1 - 1/6} = \dfrac{5/6}{5/6} = 1$.
So the sum equals $	an^{-1}(1) = \dfrac{\pi}{4}$.

Answer

$\dfrac{\pi}{3}$

Answer

$\dfrac{\pi}{2}$

Answer

$\dfrac{\pi}{6}$

If $\tan^{-1}\left(\dfrac{1}{2}\right) + \tan^{-1}\left(\dfrac{1}{3}\right) = ?$