First slide

Properties of ITF

Question

The value of x for which $\sin [\cot^{- 1} (1 + x)] = \cos (\tan^{- 1} x)$ is

Moderate

A

$\frac{1}{2}$
B

1
C

0
D

$- \frac{1}{2}$

Solution

We have, sin (cot^–1 (x + 1)) = cos(tan^–1 x)
$\Rightarrow \cos (\frac{π}{2} - \cot^{- 1} (x + 1)) = \cos (\tan^{- 1} x)$
$\Rightarrow \frac{π}{2} - \cot^{- 1} (x + 1) = 2 nπ \pm \tan^{- 1} x$
Put $n = 0 \Rightarrow \frac{π}{2} - \cot^{- 1} (x + 1) = \pm \tan^{- 1} x = \tan^{- 1} (\pm x)$
$\begin{array}{l} \Rightarrow \frac{π}{2} = \tan^{- 1} (+ x) + \cot^{- 1} (x + 1) \\ \Rightarrow x + 1 = \pm x \Rightarrow 2 x + 1 = 0 \\ ∴ x = - \frac{1}{2} \end{array}$

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