First slide

Properties of ITF

Question

If $0 \leq x \leq 1$ , then
$\tan \{\frac{1}{2} \sin^{- 1} \frac{2 x}{1 + x^{2}} + \frac{1}{2} \cos^{- 1} \frac{1 - x^{2}}{1 + x^{2}}\}$ is equal to

Moderate

A

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

0
C

$\frac{2 x}{1 + x^{2}}$
D

$x$

Solution

We know that
$\sin^{- 1} (\frac{2 x}{1 + x^{2}}) = 2 \tan^{- 1} x,$ if $- 1 \leq x \leq 1$
and,
$\begin{array}{l} \cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) = 2 \tan^{- 1} x, if 0 \leq x < \infty \\ ∴ \tan \{\frac{1}{2} \sin^{- 1} \frac{2 x}{1 + x^{2}} + \frac{1}{2} \cos^{- 1} \frac{1 - x^{2}}{1 + x^{2}}\} \\ = \tan (2 \tan^{- 1} x) for 0 \leq x \leq 1 \\ = \tan (\tan^{- 1} \frac{2 x}{1 - x^{2}}) = \frac{2 x}{1 - x^{2}} \end{array}$

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