First slide

Properties of ITF

Question

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

Moderate

A

$3 \tan^{- 1} (\frac{x}{a})$
B

$2 \tan^{- 1} (\frac{x}{a})$
C

$3 \tan^{- 1} (\frac{a}{x})$
D

None of these

Solution

Let x = a tan $θ$
$\begin{array}{ll} \Rightarrow \frac{x}{a} = \tan θ \\ \Rightarrow θ = \tan^{- 1} (\frac{x}{a}) \end{array}$
$∴ \tan^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 {ax}^{2}}) = \tan^{- 1} [\frac{3 a^{2} (atan θ) - (atan θ)^{3}}{a^{3} - 3 a (atan θ)^{2}}] \begin{array}{l} = \tan^{- 1} [\frac{a^{3} (3 \tan θ - \tan^{3} θ)}{a^{3} (1 - 3 \tan^{2} θ)}] \\ = \tan^{- 1} (\frac{3 \tan θ - \tan^{3} θ}{1 - 3 \tan^{2} θ}) \\ = \tan^{- 1} (\tan 3 θ) \\ = 3 θ = 3 \tan^{- 1} (\frac{x}{a}) [from Eq. (i)] \end{array}$

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