Question

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

Answer 1

${\tan }^{-1}\frac{x}{\sqrt{a^2-x^2}}$

Let $x=a\sin \theta$

$\frac{x}{a}=\sin \theta ⟹\theta ={\sin }^{-1}\frac{x}{a}$

Therefore,

=${\tan }^{-1}\frac{x}{\sqrt{a^2-x^2}}$

=${\tan }^{-1}\left(\frac{asin\theta }{\sqrt{a^2-a^{2}si{n}^2\theta }}\right)$ .............(since $x=asin\theta$ )

=${\tan }^{-1}\left(\frac{asin\theta }{a\sqrt{1-si{n}^2\theta }}\right)$

=${\tan }^{-1}\left(\frac{asin\theta }{a\sqrt{co{s}^2\theta }}\right)$

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

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

$=\theta$

$={\sin }^{-1}\left(\frac{x}{a}\right)$.............(since $\theta =\frac{x}{a}$)