First slide

Properties of ITF

Question

The value of $x$ for which $S i n [C o t^{- 1} (1 + x)] = C o s [T a n^{- 1} x]$ is

Moderate

A

$\frac{1}{2}$
B

1
C

0
D

$- \frac{1}{2}$

Solution

$W e h a v e \sin [c o t^{- 1} (1 + x)] = \cos (\tan^{- 1} x) \Rightarrow \sin [\sin^{- 1} (\frac{1}{\sqrt{1 + {(1 + x)}^{2}}})] = \cos [\cos^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})] \Rightarrow \frac{1}{\sqrt{1 + {(1 + x)}^{2}}} = \frac{1}{\sqrt{1 + x^{2}}} \Rightarrow 1 + x^{2} = 1 + {(1 + x)}^{2} \Rightarrow x = - \frac{1}{2}$

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