(1) \[2{{\sin }^{-1}}x={{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}}})\], If \[-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}\]\[\]
(2) \[2{{\sin }^{-1}}x=\pi -{{\sin }^{-1}}2x\sqrt{1-{{x}^{2}}}\], If \[\frac{1}{\sqrt{2}}\le x\le 1\]
(3) \[2{{\sin }^{-1}}x=-\pi -{{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}})}\], If \[-1\le x\le \frac{-1}{\sqrt{2}}\]
(4) \[3{{\sin }^{-1}}x={{\sin }^{-1}}(3x-4{{x}^{3}}),\] If \[\frac{-1}{2}\le x\le \frac{1}{2}\]
(5) \[3{{\sin }^{-1}}x=\pi -{{\sin }^{-1}}(3x-4{{x}^{3}})\], If \[\frac{1}{2}<x\le 1\]
(6) \[3{{\sin }^{-1}}x=-\pi -{{\sin }^{-1}}(3x-4{{x}^{3}}),\] If \[-1\le x<-\frac{1}{2}\]
(7) \[2{{\cos }^{-1}}x={{\cos }^{-1}}(2{{x}^{2}}-1)\], If \[0\le x\le 1\]
(8) \[2{{\cos }^{-1}}x=2\pi -{{\cos }^{-1}}(2{{x}^{2}}-1)\], If \[-1\le x\le 0\]
(9) \[3{{\cos }^{-1}}x={{\cos }^{-1}}(4{{x}^{3}}-3x)\], If \[\frac{1}{2}\le x\le 1\]
(10) \[3{{\cos }^{-1}}x=2\pi -{{\cos }^{-1}}(4{{x}^{3}}-3x),\] If \[-\frac{1}{2}\le x\le \frac{1}{2}\]
(11) \[3{{\cos }^{-1}}x=2\pi +{{\cos }^{-1}}(4{{x}^{3}}-3x),\] If \[-1\le x\le -\frac{1}{2}\]
(12) \[2{{\tan }^{-1}}x={{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)\], If \[-1<x\le 1\]
(13) \[2{{\tan }^{-1}}x=\pi +{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)\] , If \[x>1\]
(14) \[2{{\tan }^{-1}}x=-\pi +{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)\], If \[x<-1\]
(15) \[2{{\tan }^{-1}}x={{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] , If \[-1\le x\le 1\]
(16) \[2{{\tan }^{-1}}x=\pi -{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] , If \[x>1\]
(17) \[2{{\tan }^{-1}}x=-\pi -{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] , If \[x<-1\]
(18) \[2{{\tan }^{-1}}x={{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\], If \[0\le x<\infty \]
(19) \[2{{\tan }^{-1}}x=-{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\] , If \[-\infty <x\le 0\]
(20) \[3{{\tan }^{-1}}x={{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\], If \[\frac{-1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\]
(21) \[3{{\tan }^{-1}}x=\pi +{{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\] , If \[x>\frac{1}{\sqrt{3}}\]
(22) \[3{{\tan }^{-1}}x=-\pi +{{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\] , If \[x<-\frac{1}{\sqrt{3}}\]
(23) \[{{\tan }^{-1}}\left[ \frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right]={{\sin }^{-1}}\frac{x}{a}\]
(24) \[{{\tan }^{-1}}\left[ \frac{3{{a}^{2}}x-{{x}^{3}}}{a({{a}^{2}}-3{{x}^{2}})} \right]=3{{\tan }^{-1}}\frac{x}{a}\]
(25) \[{{\tan }^{-1}}\left[ \frac{\sqrt{1+{{x}^{2}}}+\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}-\sqrt{1-{{x}^{2}}}} \right]=\frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}{{x}^{2}}\]
(26) \[{{\tan }^{-1}}\sqrt{\frac{1-x}{1+x}}=\frac{1}{2}{{\cos }^{-1}}x\]