Write the following functions in the simplest form:

    9. \tan^{-1} \frac{x}{\sqrt{a^2 - x^2}}, \;\; |x| < a

Answers (1)

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

Take x = a\sin \theta   or

  \theta = \sin^{-1}\left ( \frac{x}{a} \right )   and putting it in the equation above;

\tan^{-1} \frac{a\sin \theta}{\sqrt{a^2 - (a\sin \theta)^2}}

=\tan^{-1} \frac{a\sin \theta}{a\sqrt{1 - \sin^2 \theta}}

=\tan^{-1} \left ( \frac{\sin \theta}{\sqrt{\cos^2 \theta}} \right ) = \tan^{-1} \left ( \frac{\sin \theta}{{\cos \theta}} \right )

  =\tan^{-1}\left ( \tan \theta \right )

 =\theta = \sin^{-1}\left ( \frac{x}{a} \right )  is the simplest form.

 

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