First slide

Properties of ITF

Question

If $- \infty < x \leq 0,$ then $\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$ equals

Moderate

A

$2 \tan^{- 1} x$
B

$- 2 \tan^{- 1} x$
C

$π - 2 \tan^{- 1} x$
D

$π + 2 \tan^{- 1} x$

Solution

Let $\tan^{- 1} x = θ .$ Then, $x = \tan θ$
Now,
$- \infty < x \leq 0 \Rightarrow - \infty < \tan θ \leq 0 \Rightarrow - \frac{π}{2} < θ \leq 0 \Rightarrow - π < 2 θ \leq 0$
$\begin{array}{l} ∴ \cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) \\ = \cos^{- 1} (\cos 2 θ) \end{array}$
$= \cos^{- 1} (\cos (- 2 θ))$
$= - 2 θ = - 2 \tan^{- 1} x [∵ 0 \leq - 2 θ < π]$

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