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Write the following function in the simp...

Write the following function in the simplest form: `tan^(-1)((3a^2x-x^3)/(a^3-3a x^2)), a >0;(-a)/(sqrt(3))lt=xlt=a/(sqrt(3))`

Text Solution

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`tan^(-1) ((3a^(2) x- x^(3))/(a^(3) - 3ax^(2)))`
Put `x = a tan theta " or " (x)/(a) = tan theta " or " theta = tan^(-1).(x)/(a)`
We have
`". 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) ((3a^(3) tan theta - a^(3) tan^(3) theta)/(a^(3) - 3a^(3) tan^(2) theta))`
`= tan^(-1) ((3 tan theta - tan^(3) theta)/(1- 3 tan^(2) theta))`
`= tan^(-1) (tan 3 theta) = 3 theta`
`= 3 tan^(-1).(x)/(a)`
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