Correct Answer - D
Given, ` alpha = cos ^(-1) ((3)/(5)) and beta = tan ^(-1) ((1)/(3))`
where, ` 0 lt alpha, beta lt (pi)/(2)`
Clearly, ` alpha = tan ^(-1) ""( 4)/(3)`
So, ` alpha - beta = tan ^(-1) ""( 4)/(3) - tan ^(-1) ""(1)/(3) = tan ^(-2) ((( 4)/(3) - (1)/(3))/( 1 + ((4)/(3) xx (1)/(3))))`
`" " [ because tan ^(-1) x - tan ^(-1) y = tan ^(-1) ""( x- y )/( 1 + xy ), if xy gt - 1 ]`
`= tan ^(-1) ""(1)/(1 + (4)/(9)) = tan ^(-1) ""(9)/( 13)`
`= sin ^(-1) ""(9)/( sqrt (9^(2) + 13^(2))) = sin ^(-1) ""(9)/(sqrt( 9^(2) + 13^(2)))= sin ^(-1) ""(9)/(sqrt ( 250))`
` = sin ^(-1) ((9)/( 5 sqrt(10)))`