JEE Main & Advanced Mathematics Inverse Trigonometric Functions Formulae for Sum, Difference of Inverse Trigonometric Function

(1) \[{{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)\]; 

If \[x>0,y>0\] and \[xy<1\]

(2) \[{{\tan }^{-1}}x+{{\tan }^{-1}}y=\pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)\];

If \[x>0,\,y>0\] and \[xy>1\]

(3) \[{{\tan }^{-1}}x+{{\tan }^{-1}}y=-\pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)\];            

If \[x<0,\,y<0\] and \[xy>1\]

(4) \[{{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right)\];                        

If \[xy>-1\]

(5) \[{{\tan }^{-1}}x-{{\tan }^{-1}}y=\pi +{{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right)\] ;                               

If \[x>0,\,y<0\] and \[xy<-1\]

(6) \[{{\tan }^{-1}}x-{{\tan }^{-1}}y=-\pi +{{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right)\];              

If \[x<0,\,y>0\] and \[xy<-1\]

(7) \[{{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z={{\tan }^{-1}}\left[ \frac{x+y+z-xyz}{1-xy-yz-zx} \right]\]

(8) \[{{\tan }^{-1}}{{x}_{1}}+{{\tan }^{-1}}{{x}_{2}}+..........+{{\tan }^{-1}}{{x}_{n}}\] \[={{\tan }^{-1}}\left[ \frac{{{S}_{1}}-{{S}_{3}}+{{S}_{5}}-...........}{1-{{S}_{2}}+{{S}_{4}}-{{S}_{6}}+........} \right]\]

where \[{{S}_{k}}\] denotes the sum of the products of \[{{x}_{1}},\,{{x}_{2}},........,{{x}_{n}}\] taken k  at a time.

(9) \[{{\cot }^{-1}}x+{{\cot }^{-1}}y={{\cot }^{-1}}\frac{xy-1}{y+x}\]

(10) \[{{\cot }^{-1}}x-{{\cot }^{-1}}y={{\cot }^{-1}}\frac{xy+1}{y-x}\]

(11) \[{{\sin }^{-1}}x+{{\sin }^{-1}}y={{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}}\}\];

If \[-1\le x,\,y\le 1\]and\[{{x}^{2}}+{{y}^{2}}\le 1\] or if \[xy<0\] and \[{{x}^{2}}+{{y}^{2}}>1\]

(12) \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=\pi -{{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}}\},\]

 If \[0<x\],\[y\le 1\] and \[{{x}^{2}}+{{y}^{2}}>1\]

(13) \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=-\pi -{{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}}\},\]

If \[-1\le x;\,y<0\] and \[{{x}^{2}}+{{y}^{2}}>1\]

(14) \[{{\sin }^{-1}}x-{{\sin }^{-1}}y={{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}}\},\]

If \[-1\le x;\,y\le 1\]and\[{{x}^{2}}+{{y}^{2}}\le 1\]if or \[xy>0\] and\[{{x}^{2}}+{{y}^{2}}>1\].

(15) \[{{\sin }^{-1}}x-{{\sin }^{-1}}y=\pi -{{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}}\},\]

If \[0<x\le 1,\,-1\le y<0\] and \[{{x}^{2}}+{{y}^{2}}>1\].

(16) \[{{\sin }^{-1}}x-{{\sin }^{-1}}y=-\pi -{{\sin }^{-1}}\{x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}}\},\]

If \[-1\le x<0,\,0<y\le 1\] and \[{{x}^{2}}+{{y}^{2}}>1\].

(17) \[{{\cos }^{-1}}x+{{\cos }^{-1}}y={{\cos }^{-1}}\{xy-\sqrt{1-{{x}^{2}}}.\sqrt{1-{{y}^{2}}}\}\],

If \[-1\le x,\,y\le 1\] and \[x+y\ge 0\].

(18) \[{{\cos }^{-1}}x+{{\cos }^{-1}}y=2\pi -{{\cos }^{-1}}\{xy-\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}}\}\],

If \[-1\le x,\,y\le 1\] and \[x+y\le 0\]

(19) \[{{\cos }^{-1}}x-{{\cos }^{-1}}y={{\cos }^{-1}}\{xy+\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}}\},\] 

If \[-1\le x,y\le 1,\]  and \[x\le y\].

(20) \[{{\cos }^{-1}}x-{{\cos }^{-1}}y=-{{\cos }^{-1}}\{xy+\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}}\},\]

If \[-1\le y\le 0,\] \[0<x\le 1\] and \[x\ge y\].