This document contains two problems from inverse trigonometry. The first problem involves finding the values of x and y given trigonometric expressions involving tan(x) and tan(y). The second problem proves the identity x = -x + pi for x in the range (-pi, pi). Both problems are solved step-by-step using trigonometric identities and properties. The document also provides contact information for the physics help website.

1. Physics Helpline
L K Satapathy
Inverse Trigonometry 5

2. Inverse Trigonometry 5
Physics Helpline
L K Satapathy
Question :
Answer
2
1 1
2 2
11 2tan sin cos , 1, 0, 1
2 1 1
yxFind the value of x y xy
x y
  
    
  
2
1 1
2 2
11 2tan sin cos
2 1 1
yx
x y
  
 
  
1 11tan [sin (sin 2 ) cos (cos2 )]
2
  
 
tan tan1tan (2 2 ) tan( )
2 1 tan .tan 1
[ ]
x y
xy
Ans
 
   
 

     
 
2
1 1
2 2
1 tan1 2tantan sin cos
2 1 tan 1 tan

 
  
  
  
Putting x = tan and y = tan

3. Inverse Trigonometry 5
Physics Helpline
L K Satapathy
Question :
Answer
   1 cos, tan , ,
1 sin 4 2 2 2
x xProve that x
x
  
   

 
2 2
1 1
2 2
cos sin
cos 2 2tan tan
1 sin cos sin 2sin cos
2 2 2 2
x x
xLHS
x x x x x
 
 
 
      
 
 
2 2
1 1
2
cos sin cos sin
2 2 2 2tan tan
cos sincos sin 2 22 2
x x x x
x xx x
 
       
    
       
 1 1
tan tan
4 2tan tan tan
4 2 4 21 tan tan
[ ]
4 2
x
x RHx S
x

 

 
 
            
 


4. Physics Helpline
L K Satapathy
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