First slide

Properties of ITF

Question

The value of $\sin [\cot^{- 1} \{\cos (\tan^{- 1} 1)\}],$ is

Moderate

A

$\frac{2}{3}$
B

$\sqrt{\frac{2}{3}}$
C

$\frac{1}{\sqrt{2}}$
D

$\sqrt{\frac{3}{2}}$

Solution

$\sin [\cot^{- 1} \{\cos (\tan^{- 1} 1)\}]$
$\begin{array}{l} = \sin \{\cot^{- 1} (\cos \frac{π}{4})\} [∵ \tan^{- 1} 1 = \frac{π}{4}] \\ = \sin (\cot^{- 1} \frac{1}{\sqrt{2}}) \\ = \sin (\sin^{- 1} \frac{\sqrt{2}}{\sqrt{3}}) = \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}} [∵ \cot^{- 1} \frac{1}{\sqrt{2}} = \sin^{- 1} \frac{\sqrt{2}}{\sqrt{3}}] \end{array}$

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