First slide

Properties of ITF

Question

$\cos^{- 1} [\cos \{2 \cot^{- 1} (\sqrt{2} - 1)\}]$ is equal to

Easy

A

$\sqrt{2} - 1$
B

$1 - \sqrt{2}$
C

$\frac{π}{4}$
D

$\frac{3 π}{4}$

Solution

Let cot^-1( $\sqrt{2}$ -1) = 0
⇒ cotθ = $\sqrt{2}$ -1
$Now, \cos^{- 1} (\cos 2 θ) = \cos^{- 1} (\frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}) \begin{array}{l} = \cos^{- 1} (\frac{\cot^{2} θ - 1}{\cot^{2} θ + 1}) = \cos^{- 1} (\frac{2 - 2 \sqrt{2}}{4 - 2 \sqrt{2}}) \\ ∴ \cos^{- 1} \cos (2 θ) = \cos^{- 1} (- \frac{1}{\sqrt{2}}) = \frac{3 π}{4} \end{array}$

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