First slide

Introduction to ITF

Question

The range of the value of p for which the equation $\sin (\cos^{- 1} (\cos (\tan^{- 1} x))) = p$ has a solution is

Moderate

A

$(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]$
B

$[0, 1)$
C

$[\frac{1}{\sqrt{2}}, 1)$
D

$(- 1, 1)$

Solution

${sincos}^{- 1} (\cos (\tan^{- 1} x)) = p$
For $x \in R, \tan^{- 1} x \in (- π / 2, π / 2)$
$\Rightarrow \cos (\tan^{- 1} x) \in (0, 1]$
$\Rightarrow \cos^{- 1} \cos (\tan^{- 1} x) \in [0, π / 2)$
$\Rightarrow \sin [\cos^{- 1} (\cos (\tan^{- 1} x))] \in [0, 1) \Rightarrow p \in [0, 1)$

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