First slide

Properties of ITF

Question

If –1 < x < 0 then tan^–1x equals

Moderate

A

$π - \cos^{- 1} (\sqrt{1 - x^{2}})$
B

$\sin^{- 1} (\frac{x}{\sqrt{1 + x^{2}}})$
C

$- \cot^{- 1} (\frac{\sqrt{1 - x^{2}}}{x})$
D

cosec^–1x

Solution

$∵ - 1 < x < 0, ∴ - \frac{π}{4} < \tan^{- 1} x < 0 Let \tan^{- 1} x = α \Rightarrow - \frac{π}{4} < α < 0 ∴ \tan α = x, - \frac{π}{4} < α < 0 \begin{array}{l} ∴ \sin α = \frac{x}{\sqrt{1 + x^{2}}} \Rightarrow α = \sin^{- 1} \frac{x}{\sqrt{1 + x^{2}}} \\ and \cos α = \frac{1}{\sqrt{1 + x^{2}}} \Rightarrow \cos (- α) = \frac{1}{\sqrt{1 + x^{2}}}, \\ where 0 < - α < \frac{π}{4} \Rightarrow α = - \cos^{- 1} (\frac{1}{\sqrt{1 + x^{2}}}) \end{array}$

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