Write the function in the simplest form: ^ - 1 x a^2 - x^2 , | x … - Vidyadip

Question

Write the function in the simplest form: ${\tan ^{ - 1}}\frac{x}{{\sqrt {{a^2} - {x^2}} }},\left| x \right| < a$

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Answer

$$ Put $x = a\sin \theta$ so that $\theta = {\sin ^{ - 1}}\frac{x}{a}$
$ \Rightarrow {\tan ^{ - 1}}\left( {\frac{{a\sin \theta }}{{\sqrt {{a^2} - {a^2}{{\sin }^2}\theta } }}} \right)$
$ = {\tan ^{ - 1}}\left( {\frac{{a\sin \theta }}{{\sqrt {{a^2}(1 - {{\sin }^2}\theta } }}} \right)$
$ = {\tan ^{ - 1}}\left( {\frac{{a\sin \theta }}{{\sqrt {{a^2}{{\cos }^2}\theta } }}} \right)$
$ = {\tan ^{ - 1}}\left( {\frac{{a\sin \theta }}{{a\cos \theta }}} \right)$
$= {\tan ^{ - 1}}\tan \theta$
$= \theta = {\sin ^{ - 1}}\frac{x}{a}$

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