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Question: The value of \({\sin h}^{- 1}\left( \frac{x}{\sqrt{1 - x^{2}}} \right)\) is...

The value of sin⁡h−1(x1−x2){\sin h}^{- 1}\left( \frac{x}{\sqrt{1 - x^{2}}} \right) is

A

tanh⁡−1x\tanh^{- 1}x

B

coth⁡−1x\coth^{- 1}x

C

sinh⁡−1(2x)\sinh^{- 1}(2x)

D

cos⁡h−1(2x){\cos h}^{- 1}(2x)

Answer

tanh⁡−1x\tanh^{- 1}x

Explanation

Solution

Let x=tanh⁡yx = \tanh y, then x1−x2=tan⁡hysec⁡hy=sin⁡hy\frac{x}{\sqrt{1 - x^{2}}} = \frac{{\tan h}y}{\sec hy} = {\sin h}y

∴\therefore sinh⁡−1(x1−x2)=sinh⁡−1(sinh⁡y)\sinh^{- 1}\left( \frac{x}{\sqrt{1 - x^{2}}} \right) = \sinh^{- 1}(\sinh y) ⇒ y=tan⁡h−1(x)y = {\tan h}^{- 1}(x)