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

Question

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

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Answer

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

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