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The simplified form of $\tan^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 {ax}^{2}}), a > 0; \frac{- a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}} is$

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a

3tan−1⁡xa

b

2tan−1⁡xa

c

3tan−1⁡ax

d

None of these

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detailed solution

Correct option is A

Let x = a tan θ⇒ xa=tan⁡θ⇒ θ=tan−1⁡xa∴tan−1⁡3a2x−x3a3−3ax2=tan−1⁡3a2(atan⁡θ)−(atan⁡θ)3a3−3a(atan⁡θ)2 =tan−1⁡a33tan⁡θ−tan3⁡θa31−3tan2⁡θ=tan−1⁡3tan⁡θ−tan3⁡θ1−3tan2⁡θ=tan−1⁡(tan⁡3θ)=3θ=3tan−1⁡xa [from Eq. (i)]

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