Prove that:\(\tan^{-1}\left(\frac{\mathrm x+\sqrt{\mathrm x}}{1-\mathr

1.	Prove that:\(\tan^{-1}\left(\frac{\mathrm x+\sqrt{\mathrm x}}{1-\mathrm x^{3/2}}\right)\)\(=\tan^{-1}\mathrm x+ \tan^{-1}\sqrt{\mathrm x}\)
Answer» We know that, \(\tan^{-1}\left(\frac{A+B}{1-AB}\right)\)\(=\tan^{-1}A+\tan^{-1}B\) Now, taking A = x and B = √x We get, \(\tan^{-1}\mathrm x+\tan ^{-1}\sqrt{\mathrm x}\)\(=\tan^{-1}\left(\frac{\mathrm x+\sqrt{\mathrm x}}{1-\mathrm x^{3/2}}\right)\) As, x. x ^1/2 = x ^3/2 Hence, Proved.

Prove that:\(\tan^{-1}\left(\frac{\mathrm x+\sqrt{\mathrm x}}{1-\mathrm x^{3/2}}\right)\)\(=\tan^{-1}\mathrm x+ \tan^{-1}\sqrt{\mathrm x}\)

Answer»

We know that,

\(\tan^{-1}\left(\frac{A+B}{1-AB}\right)\)\(=\tan^{-1}A+\tan^{-1}B\)

Now, taking A = x and B = √x

We get,

\(\tan^{-1}\mathrm x+\tan ^{-1}\sqrt{\mathrm x}\)\(=\tan^{-1}\left(\frac{\mathrm x+\sqrt{\mathrm x}}{1-\mathrm x^{3/2}}\right)\)

As, x. x ^1/2 = x ^3/2

Hence, Proved.