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Question
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 < x < a/sqrt3`
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Solution
Put x = a tan θ
⇒ `x/a` = tan θ
⇒ θ
= `tan^(-1) x/a`
∴ `tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)) `
= `tan^(-1) ((3a^2. atan θ - a^3 tan^3 θ)/(a^3 - 3a.a^2 tan^2 θ))`
= `tan^(-1) ((3a^3 tan θ - a^3 tan^3 θ)/(a^3 - 3a^3 tan^2 θ))`
= `tan^(-1) ((3tan θ - tan^3 θ)/(1 -3tan^2 θ))`
= tan−1 (tan 3θ)
= 3θ
= `3 tan^(-1) x/a`
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