First slide

Properties of ITF

Question

If $x < - \frac{1}{\sqrt{3}},$ then $\tan^{- 1} (\frac{3 x - x^{3}}{1 - 3 x^{2}})$ equals

Moderate

A

$3 \tan^{- 1} x$
B

$- π + 3 \tan^{- 1} x$
C

$π + 3 \tan^{- 1} x$
D

none of these

Solution

Let $\tan^{- 1} x = θ .$ Then, $x = \tan θ$
Also,
$x < - \frac{1}{\sqrt{3}} \Rightarrow \tan θ < - \frac{1}{\sqrt{3}} \Rightarrow - \frac{π}{2} < θ < - \frac{π}{6}$
Now,
$\begin{array}{l} \tan^{- 1} (\frac{3 x - x^{3}}{1 - 3 x^{2}}) \\ = \tan^{- 1} (\tan 3 θ) \\ = \tan^{- 1} (\tan (π + 30)) = π + 3 θ = π + 3 \tan^{- 1} x \end{array}$

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