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

1.	Prove that:\(\tan^{-1}\left(\frac{\sqrt{\mathrm x}+\sqrt{\mathrm y}}{1-\sqrt{\mathrm {xy}}}\right)=\tan^{-1}\sqrt{\mathrm x}+tan^{-1}\sqrt{\mathrm y}\)
Answer» To Prove: \(\tan^{-1}\left(\frac{\sqrt{\mathrm x}+\sqrt{\mathrm y}}{1-\sqrt{\mathrm {xy}}}\right)=\tan^{-1}\sqrt{\mathrm x}+tan^{-1}\sqrt{\mathrm y}\) We know that, \(\tan A+\tan B = \frac{\tan A+\tan B}{1-\tan A \tan B}\) Also, \(\tan^{-1}\left(\frac{A+B}{1-AB}\right)=\tan^{-1}A + \tan ^{-1} B\) Taking A = √x and B = √y We get, \(\tan^{-1}\left(\frac{\sqrt{\mathrm x}+\sqrt{\mathrm y}}{1-\sqrt{\mathrm{xy}}}\right)\)\(=\tan^{-1}\sqrt{\mathrm x}+\tan^{-1}\sqrt{\mathrm y}\) Hence, Proved.

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

Answer»

To Prove:

\(\tan^{-1}\left(\frac{\sqrt{\mathrm x}+\sqrt{\mathrm y}}{1-\sqrt{\mathrm {xy}}}\right)=\tan^{-1}\sqrt{\mathrm x}+tan^{-1}\sqrt{\mathrm y}\)

We know that,

\(\tan A+\tan B = \frac{\tan A+\tan B}{1-\tan A \tan B}\)

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

Taking A = √x and B = √y

We get,

\(\tan^{-1}\left(\frac{\sqrt{\mathrm x}+\sqrt{\mathrm y}}{1-\sqrt{\mathrm{xy}}}\right)\)\(=\tan^{-1}\sqrt{\mathrm x}+\tan^{-1}\sqrt{\mathrm y}\)

Hence, Proved.