The value of \(\tan \left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]\) is

The value of \(\tan \left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]\) is to be found out. Let \(\quad \cos ^{-1}\left(\frac{4}{5}\right)=\theta \Rightarrow \cos \theta=\frac{4}{5} \Rightarrow \sin \theta=\frac{3}{5}\)
\(\therefore \tan \theta=\frac{3}{4} \Rightarrow \theta=\tan ^{-1}\left(\frac{3}{4}\right)\)
Hence given expression.
\(=\tan \left[\tan ^{-1}\left(\frac{3}{4}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]\) \(=\tan \left[\tan ^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\left(\frac{3}{4}\right)\left(\frac{2}{3}\right)}\right)\right] \)
\( =\tan \left[\tan ^{-1}\left(\frac{17}{6}\right)\right]=\frac{17}{6} \)