( ^ - 1 (4)/(5) + ^ - 1 (4)/(3) + ^ - 1 (1)/(9) - ^ - 1 (1)/(7)) =

\sin \left(\tan^{- 1}\dfrac{4}{5} + \tan^{- 1}\dfrac{4}{3} + \tan^{- 1}\dfrac{1}{9} - \tan^{- 1}\dfrac{1}{7}\right)$</p><p>$= \sin \left(\tan^{- 1}\dfrac{4}{5} - \tan^{- 1}\dfrac{1}{7} + \tan^{- 1}\dfrac{4}{3} + \tan^{- 1}\dfrac{1}{9}\right)$</p><p>Now,~$\tan^{- 1}\dfrac{4}{5} - \tan^{- 1}\dfrac{1}{7} = p \left(say\right)$</p><p>$⇒ p = \tan^{- 1}\left(\dfrac{\dfrac{4}{5} - \dfrac{1}{7}}{1 + \dfrac{4}{5} \cdot \dfrac{1}{7}}\right) = \tan^{- 1} $\left(\dfrac{23}{39}\right)</p><p>Also,~$\tan^{- 1}\dfrac{4}{3} + \tan^{-1}\dfrac{1}{9} = q \left(say\right)$</p><p>$⇒ q = \tan^{- 1}\left(\dfrac{\dfrac{4}{3} + \dfrac{1}{9}}{1 - \dfrac{4}{3} \cdot \dfrac{1}{9}}\right) = \tan^{- 1}\left(\dfrac{39}{23}\right)

$⇒ q = \cot^{- 1} \left(\dfrac{23}{39}\right) = \dfrac{π}{2} - \tan^{- 1} \left(\dfrac{23}{39}\right)$

So,~\sin \left(p + q\right) = sin$\left(\dfrac{π}{2}\right) =1$