First slide

Properties of ITF

Question

$2 \tan^{- 1} (- 2)$ is equal to

Moderate

A

$- \cos^{- 1} (\frac{- 3}{5})$
B

$- π + \cos^{- 1} \frac{3}{5}$
C

$- \frac{π}{2} + \tan^{- 1} (- \frac{3}{4})$
D

$- π + \cot^{- 1} (- \frac{3}{4})$

Solution

Let $\tan^{- 1} (- 2) = θ or \tan θ = - 2$
$\begin{matrix} \Rightarrow θ \in (- π / 2, 0) or 2 θ \in (- π, 0) \\ \cos (- 2 θ) = \cos 2 θ = \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ} = \frac{- 3}{5} \\ or - 2 θ = \cos^{- 1} (\frac{- 3}{5}) = π - \cos^{- 1} \frac{3}{5} \\ or 2 θ = - π + \cos^{- 1} \frac{3}{5} = - π + \tan^{- 1} \frac{4}{3} \\ = - π + \cot^{- 1} \frac{3}{4} = - π + \frac{π}{2} - \tan^{- 1} \frac{3}{4} \\ = - \frac{π}{2} - \tan^{- 1} \frac{3}{4} = - \frac{π}{2} + \tan^{- 1} (- \frac{3}{4}) \end{matrix}$

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