`tan^(-1)((3a^(2)x-x^(3))/(a^(3)-3ax^(2))),agt0 , = (a)/(sqrt(3)) lt x lt (a)/(sqrt(3))` को सरलतम रूप में लिखे |

Explanation

`x = a tan theta` रखें | `because - (a)/(sqrt(3)) lt x lt (a)/(sqrt(3))` तथा `a gt 0`
`therefore - (a)/(sqrt(3)) lt a tan theta lt (a)/(sqrt(3)) rArr - (1)/(sqrt(3)) lt tan theta lt (1)/(sqrt(3))`
`rArr - (pi)/(3) lt theta lt (pi)/(3)" "...(1)`
`because x = a tan theta therefore tan theta = (x)/(a) rArr theta = tan^(-1)""(x)/(a)" "...(2)`
अब, `tan^(-1)[(3a^(2)x - x^(3))/(a^(3) - 3ax^(2))] = tan^(-1)[(3a^(2)a tan theta - a^(3)tan^(3)theta)/(a^(3)-3a a^(2)tan^(2)theta)]`
` = tan^(-1)[(a^(2)(3 tan theta-tan^(3)theta))/(a^(3)(1-3tan^(2)theta))] = tan^(-1)(tan 3theta)`
`= (3theta) = 3 tan^(-1)""(x)/(a)" "`[(2) से]