If we consider only the principle values of the inverse trigonometric functions then the value of $$\tan \left( {{{\cos }^{ - 1}}\frac{1}{{5\sqrt 2 }} - {{\sin }^{ - 1}}\frac{4}{{\sqrt {17} }}} \right)$$      is

Solution :
$$\eqalign{ & \tan \left[ {{{\cos }^{ - 1}}\frac{1}{{5\sqrt 2 }} - {{\sin }^{ - 1}}\frac{4}{{\sqrt {17} }}} \right] \cr & = \tan \left[ {{{\tan }^{ - 1}}7 - {{\tan }^{ - 1}}4} \right] \cr & = \tan \left( {{{\tan }^{ - 1}}\left( {\frac{3}{{29}}} \right)} \right) \cr & = \frac{3}{{29}} \cr} $$