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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)/sqrt(3) < x < a/sqrt(3)`
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Solution
`tan^(-1) ((3a^2x - x^3)/(a^3 - 3ax^2))`
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 · a tan 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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