The value of 2 ^ -1 (4)/(5) + 2 ^ -1 (5)/(13) + 2 ^ -1 (16)/(65) is...

2 \left[\sin^{- 1} $\dfrac{4}{5} + \sin^{- 1}$ $\dfrac{5}{13} + \sin^{- 1}$ \dfrac{16}{65}\right]<br />= 2\sin^{- 1} \left(\dfrac{4}{5}\sqrt{1 - {\left(\dfrac{5}{13}\right)}^{2}} + \dfrac{5}{13}\sqrt{1 - {\left(\dfrac{4}{5}\right)}^{2}}\right) +2\sin^{- 1} \dfrac{16}{65}
$= 2\sin^{- 1} \left(\dfrac{48}{65} + \dfrac{15}{65}\right) + 2\sin^{- 1} \left(\dfrac{16}{65}\right)$
= 2 \left[\sin^{- 1} \left(\dfrac{63}{65}\right) + \sin^{- 1} \left(\dfrac{16}{65}\right)\right]
=2\left[\cos^{- 1}\left(\dfrac{16}{65}\right) + \sin^{- 1} \left(\dfrac{16}{65}\right)\right]\left(∵ \sin^{- 1} x = \cos^{- 1}\sqrt{1 - x^{2}}\right)
$= \pi$