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Simplificación de expresiones/ Descomposición en fracciones simples/ ctg(x)*((tg(x)+3*ctg(y))/(3*tg(y)+ctg(x)))

¿Cómo vas a descomponer esta ctg(x)*((tg(x)+3*ctg(y))/(3*tg(y)+ctg(x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
       tan(x) + 3*cot(y)
cot(x)*-----------------
       3*tan(y) + cot(x)
$$\frac{\tan{\left(x \right)} + 3 \cot{\left(y \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}} \cot{\left(x \right)}$$ 
cot(x)*((tan(x) + 3*cot(y))/(3*tan(y) + cot(x)))
Simplificación general [src] 
    3              
  ------ + tan(x)  
  tan(y)           
-------------------
1 + 3*tan(x)*tan(y)
$$\frac{\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}}{3 \tan{\left(x \right)} \tan{\left(y \right)} + 1}$$ 
(3/tan(y) + tan(x))/(1 + 3*tan(x)*tan(y))
Respuesta numérica [src] 
(3.0*cot(y) + tan(x))*cot(x)/(3.0*tan(y) + cot(x))
(3.0*cot(y) + tan(x))*cot(x)/(3.0*tan(y) + cot(x))
Denominador común [src] 
           3*tan(x)*tan(y) + 9*cot(y)*tan(y)         
3*cot(y) - --------------------------------- + tan(x)
                   3*tan(y) + cot(x)
$$- \frac{3 \tan{\left(x \right)} \tan{\left(y \right)} + 9 \tan{\left(y \right)} \cot{\left(y \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}} + \tan{\left(x \right)} + 3 \cot{\left(y \right)}$$ 
3*cot(y) - (3*tan(x)*tan(y) + 9*cot(y)*tan(y))/(3*tan(y) + cot(x)) + tan(x)
Denominador racional [src] 
(3*cot(y) + tan(x))*cot(x)
--------------------------
    3*tan(y) + cot(x)
$$\frac{\left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)/(3*tan(y) + cot(x))
Combinatoria [src] 
(3*cot(y) + tan(x))*cot(x)
--------------------------
    3*tan(y) + cot(x)
$$\frac{\left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)/(3*tan(y) + cot(x))
Unión de expresiones racionales [src] 
(3*cot(y) + tan(x))*cot(x)
--------------------------
    3*tan(y) + cot(x)
$$\frac{\left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)/(3*tan(y) + cot(x))
Abrimos la expresión [src] 
  cot(x)*tan(x)      3*cot(x)*cot(y) 
----------------- + -----------------
3*tan(y) + cot(x)   3*tan(y) + cot(x)
$$\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}} + \frac{3 \cot{\left(x \right)} \cot{\left(y \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
cot(x)*tan(x)/(3*tan(y) + cot(x)) + 3*cot(x)*cot(y)/(3*tan(y) + cot(x))
Compilar la expresión [src] 
(3*cot(y) + tan(x))*cot(x)
--------------------------
    3*tan(y) + cot(x)
$$\frac{\left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)/(3*tan(y) + cot(x))
Potencias [src] 
/             /   I*x    -I*x\\       
|           I*\- e    + e    /|       
|3*cot(y) + ------------------|*cot(x)
|               I*x    -I*x   |       
\              e    + e       /       
--------------------------------------
        /   I*y    -I*y\              
    3*I*\- e    + e    /              
    -------------------- + cot(x)     
         I*y    -I*y                  
        e    + e
$$\frac{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{\frac{3 i \left(- e^{i y} + e^{- i y}\right)}{e^{i y} + e^{- i y}} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)
--------------------------
    3*tan(y) + cot(x)
$$\frac{\left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right) \cot{\left(x \right)}}{3 \tan{\left(y \right)} + \cot{\left(x \right)}}$$ 
(3*cot(y) + tan(x))*cot(x)/(3*tan(y) + cot(x))
Parte trigonométrica [src] 
                   /    pi\
              3*sec|y - --|
   sec(x)          \    2 /
----------- + -------------
   /    pi\       sec(y)   
sec|x - --|                
   \    2 /                
---------------------------
        3*sec(x)*sec(y)    
1 + -----------------------
       /    pi\    /    pi\
    sec|x - --|*sec|y - --|
       \    2 /    \    2 /
$$\frac{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}}}{\frac{3 \sec{\left(x \right)} \sec{\left(y \right)}}{\sec{\left(x - \frac{\pi}{2} \right)} \sec{\left(y - \frac{\pi}{2} \right)}} + 1}$$ 
    3              
  ------ + tan(x)  
  tan(y)           
-------------------
1 + 3*tan(x)*tan(y)
$$\frac{\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}}{3 \tan{\left(x \right)} \tan{\left(y \right)} + 1}$$ 
/sec(x)   3*csc(y)\       
|------ + --------|*csc(x)
\csc(x)    sec(y) /       
--------------------------
/csc(x)   3*sec(y)\       
|------ + --------|*sec(x)
\sec(x)    csc(y) /
$$\frac{\left(\frac{3 \csc{\left(y \right)}}{\sec{\left(y \right)}} + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}{\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{3 \sec{\left(y \right)}}{\csc{\left(y \right)}}\right) \sec{\left(x \right)}}$$ 
     /pi    \                
  csc|-- - x|                
     \2     /     3*csc(y)   
  ----------- + -----------  
     csc(x)        /pi    \  
                csc|-- - y|  
                   \2     /  
-----------------------------
         /pi    \    /pi    \
    3*csc|-- - x|*csc|-- - y|
         \2     /    \2     /
1 + -------------------------
          csc(x)*csc(y)
$$\frac{\frac{3 \csc{\left(y \right)}}{\csc{\left(- y + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{1 + \frac{3 \csc{\left(- x + \frac{\pi}{2} \right)} \csc{\left(- y + \frac{\pi}{2} \right)}}{\csc{\left(x \right)} \csc{\left(y \right)}}}$$ 
/  1              \       
|------ + 3*cot(y)|*cot(x)
\cot(x)           /       
--------------------------
       3                  
     ------ + cot(x)      
     cot(y)
$$\frac{\left(3 \cot{\left(y \right)} + \frac{1}{\cot{\left(x \right)}}\right) \cot{\left(x \right)}}{\cot{\left(x \right)} + \frac{3}{\cot{\left(y \right)}}}$$ 
/                   /    pi\\            
|              3*sec|y - --||            
|   sec(x)          \    2 /|    /    pi\
|----------- + -------------|*sec|x - --|
|   /    pi\       sec(y)   |    \    2 /
|sec|x - --|                |            
\   \    2 /                /            
-----------------------------------------
    /   /    pi\              \          
    |sec|x - --|              |          
    |   \    2 /     3*sec(y) |          
    |----------- + -----------|*sec(x)   
    |   sec(x)        /    pi\|          
    |              sec|y - --||          
    \                 \    2 //
$$\frac{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}{\left(\frac{3 \sec{\left(y \right)}}{\sec{\left(y - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$ 
/     2                \         
|2*sin (x)   3*sin(2*y)|         
|--------- + ----------|*sin(2*x)
| sin(2*x)        2    |         
\            2*sin (y) /         
---------------------------------
  /                 2   \        
  | sin(2*x)   6*sin (y)|    2   
2*|--------- + ---------|*sin (x)
  |     2       sin(2*y)|        
  \2*sin (x)            /
$$\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{3 \sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}}\right) \sin{\left(2 x \right)}}{2 \left(\frac{6 \sin^{2}{\left(y \right)}}{\sin{\left(2 y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}$$ 
  1              
------ + 3*cot(y)
cot(x)           
-----------------
          3      
1 + -------------
    cot(x)*cot(y)
$$\frac{3 \cot{\left(y \right)} + \frac{1}{\cot{\left(x \right)}}}{1 + \frac{3}{\cot{\left(x \right)} \cot{\left(y \right)}}}$$ 
    /   /    pi\              \          
    |cos|x - --|              |          
    |   \    2 /     3*cos(y) |          
    |----------- + -----------|*cos(x)   
    |   cos(x)        /    pi\|          
    |              cos|y - --||          
    \                 \    2 //          
-----------------------------------------
/                   /    pi\\            
|              3*cos|y - --||            
|   cos(x)          \    2 /|    /    pi\
|----------- + -------------|*cos|x - --|
|   /    pi\       cos(y)   |    \    2 /
|cos|x - --|                |            
\   \    2 /                /
$$\frac{\left(\frac{3 \cos{\left(y \right)}}{\cos{\left(y - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos{\left(x \right)}}{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \cos{\left(y - \frac{\pi}{2} \right)}}{\cos{\left(y \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}$$ 
       3                  
     ------ + tan(x)      
     tan(y)               
--------------------------
/  1              \       
|------ + 3*tan(y)|*tan(x)
\tan(x)           /
$$\frac{\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}}{\left(3 \tan{\left(y \right)} + \frac{1}{\tan{\left(x \right)}}\right) \tan{\left(x \right)}}$$ 
 3*cot(y) + tan(x) 
-------------------
1 + 3*tan(x)*tan(y)
$$\frac{\tan{\left(x \right)} + 3 \cot{\left(y \right)}}{3 \tan{\left(x \right)} \tan{\left(y \right)} + 1}$$ 
     /    pi\                
  cos|x - --|                
     \    2 /     3*cos(y)   
  ----------- + -----------  
     cos(x)        /    pi\  
                cos|y - --|  
                   \    2 /  
-----------------------------
         /    pi\    /    pi\
    3*cos|x - --|*cos|y - --|
         \    2 /    \    2 /
1 + -------------------------
          cos(x)*cos(y)
$$\frac{\frac{3 \cos{\left(y \right)}}{\cos{\left(y - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{1 + \frac{3 \cos{\left(x - \frac{\pi}{2} \right)} \cos{\left(y - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} \cos{\left(y \right)}}}$$ 
    /   /pi    \              \          
    |csc|-- - x|              |          
    |   \2     /     3*csc(y) |          
    |----------- + -----------|*csc(x)   
    |   csc(x)        /pi    \|          
    |              csc|-- - y||          
    \                 \2     //          
-----------------------------------------
/                   /pi    \\            
|              3*csc|-- - y||            
|   csc(x)          \2     /|    /pi    \
|----------- + -------------|*csc|-- - x|
|   /pi    \       csc(y)   |    \2     /
|csc|-- - x|                |            
\   \2     /                /
$$\frac{\left(\frac{3 \csc{\left(y \right)}}{\csc{\left(- y + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{3 \csc{\left(- y + \frac{\pi}{2} \right)}}{\csc{\left(y \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
/sin(x)   3*cos(y)\       
|------ + --------|*cos(x)
\cos(x)    sin(y) /       
--------------------------
/cos(x)   3*sin(y)\       
|------ + --------|*sin(x)
\sin(x)    cos(y) /
$$\frac{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \cos{\left(y \right)}}{\sin{\left(y \right)}}\right) \cos{\left(x \right)}}{\left(\frac{3 \sin{\left(y \right)}}{\cos{\left(y \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin{\left(x \right)}}$$ 
     2                
2*sin (x)   3*sin(2*y)
--------- + ----------
 sin(2*x)        2    
            2*sin (y) 
----------------------
          2       2   
    12*sin (x)*sin (y)
1 + ------------------
    sin(2*x)*sin(2*y)
$$\frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{3 \sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}}}{\frac{12 \sin^{2}{\left(x \right)} \sin^{2}{\left(y \right)}}{\sin{\left(2 x \right)} \sin{\left(2 y \right)}} + 1}$$ 
(2*sin(x)^2/sin(2*x) + 3*sin(2*y)/(2*sin(y)^2))/(1 + 12*sin(x)^2*sin(y)^2/(sin(2*x)*sin(2*y)))
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