Formulae for Sum and Difference of two Tan Inverse functions with proof of the results and Relationship between Sine Inverse , Cos Inverse and tan Inverse functions

1. Physics Helpline
L K Satapathy
Inverse Trigonometry 2

2. Inverse Trigonometry 2
Physics Helpline
L K Satapathy
1 1
: tan tanProof Put x and y  
 
tan tanx and y   
tan tan
: tan( )
1 tan tan
Use
 
 
 

 

1 tan tan
tan
1 tan tan
 
 
 
  
     
1 1 1
tan tan tan
1
x y
x y
xy
    
     
1 1 1
1. tan tan tan
1
x y
x y
xy
    
    

3. Physics Helpline
L K Satapathy
1 1
: tan tanProof Put x and y  
 
tan tanx and y   
tan tan
: tan( )
1 tan tan
Use
 
 
 

 

1 tan tan
tan
1 tan tan
 
 
 
  
     
1 1 1
tan tan tan
1
x y
x y
xy
    
     
1 1 1
2. tan tan tan
1
x y
x y
xy
    
    
Inverse Trigonometry 2

4. Physics Helpline
L K Satapathy
1
: tanProof Put x 

tanx  
2
2tan: tan 2
1 tan
Use 



1
2
2tan2 tan
1 tan


     
 
1 1
2
22tan tan
1
xx
x
      
 
1 1
2
23. 2tan tan
1
xx
x
     
 
Inverse Trigonometry 2

5. Physics Helpline
L K Satapathy
1
: tanProof Put x 

tanx  
2
2tan: sin 2
1 tan
Use 



1
2
2tan2 sin
1 tan


     
 
1 1
2
22tan sin
1
xx
x
      
 
1 1
2
24. 2tan sin
1
xx
x
     
 
sin2 2sin cos  
2sin2 cos
cos
 


2
2tan
sec



2
2tan
1 tan




2
2tan cos 
Recall
Inverse Trigonometry 2

6. Physics Helpline
L K Satapathy
1
: tanProof Put x 

tanx  
2
2
1 tan: cos2
1 tan
Use 



2
1
2
1 tan2 cos
1 tan


     
 
2
1 1
2
12tan cos
1
xx
x
      
 
2
1 1
2
15. 2tan cos
1
xx
x
     
 
2 2
cos2 cos sin   
2 2 2
2
sec (cos sin )
sec
  



2
2
1 tan
sec



2
2
1 tan
1 tan




Recall
Inverse Trigonometry 2

7. Physics Helpline
L K Satapathy
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