Write the following functions in the simplest form:

    6. \tan^{-1} \frac{1}{\sqrt{x^2 -1}},\;\; |x| > 1

Answers (1)

Given that \tan^{-1} \frac{1}{\sqrt{x^2 -1}},\;\; |x| > 1

Take x =cosec\ \Theta  or  \Theta = cosec ^{-1}x

\therefore tan^{-1}\frac{1}{\sqrt{x^2-1}}

 =tan^{-1} \frac{1}{\sqrt{cosec^2 \Theta -1}}

  =tan^{-1}(\frac{1}{\cot \Theta})

 =tan^{-1}(\tan \Theta)  =  \Theta

cosec^{-1}x

=\frac{\pi}{2}- \sec^{-1}x        [\because cosec^{-1}x + \sec^{-1}x = \frac{\pi}{2}]

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