Write the function in the simplest form: tan | vedclass

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Question

(D) Given expression: $	an ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$Let $x = a \sin 	heta$. Then $\frac{x}{a} = \sin 	heta$,which implies $	heta = \sin

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$\sin ^{-1} \frac{x}{a}$

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(D) Given expression: $	an ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$Let $x = a \sin 	heta$. Then $\frac{x}{a} = \sin 	heta$,which implies $	heta = \sin ^{-1} \left(\frac{x}{a}ight)$.Substituting $x = a \sin 	heta$ into the expression:$= 	an ^{-1} \left(\frac{a \sin 	heta}{\sqrt{a^{2} - a^{2} \sin ^{2} 	heta}}ight)$$= 	an ^{-1} \left(\frac{a \sin 	heta}{\sqrt{a^{2}(1 - \sin ^{2} 	heta)}}ight)$$= 	an ^{-1} \left(\frac{a \sin 	heta}{a \cos 	heta}ight)$$= 	an ^{-1} (	an 	heta)$$= 	heta = \sin ^{-1} \left(\frac{x}{a}ight)$

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

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