Sr Examen

¿Cómo vas a descomponer esta tan(pi/(4+a))+tan(pi/(4-a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src]
   /  pi \      /  pi \
tan|-----| + tan|-----|
   \4 + a/      \4 - a/
$$\tan{\left(\frac{\pi}{4 - a} \right)} + \tan{\left(\frac{\pi}{a + 4} \right)}$$
tan(pi/(4 + a)) + tan(pi/(4 - a))
Simplificación general [src]
     /  pi  \      /  pi \
- tan|------| + tan|-----|
     \-4 + a/      \4 + a/
$$- \tan{\left(\frac{\pi}{a - 4} \right)} + \tan{\left(\frac{\pi}{a + 4} \right)}$$
-tan(pi/(-4 + a)) + tan(pi/(4 + a))
Respuesta numérica [src]
tan(pi/(4 + a)) + tan(pi/(4 - a))
tan(pi/(4 + a)) + tan(pi/(4 - a))
Potencias [src]
  /    pi*I    -pi*I \     /    pi*I    -pi*I \
  |   -----    ------|     |   -----    ------|
  |   4 + a    4 + a |     |   4 - a    4 - a |
I*\- e      + e      /   I*\- e      + e      /
---------------------- + ----------------------
     pi*I    -pi*I            pi*I    -pi*I    
    -----    ------          -----    ------   
    4 + a    4 + a           4 - a    4 - a    
   e      + e               e      + e         
$$\frac{i \left(- e^{\frac{i \pi}{4 - a}} + e^{- \frac{i \pi}{4 - a}}\right)}{e^{\frac{i \pi}{4 - a}} + e^{- \frac{i \pi}{4 - a}}} + \frac{i \left(- e^{\frac{i \pi}{a + 4}} + e^{- \frac{i \pi}{a + 4}}\right)}{e^{\frac{i \pi}{a + 4}} + e^{- \frac{i \pi}{a + 4}}}$$
i*(-exp(pi*i/(4 + a)) + exp(-pi*i/(4 + a)))/(exp(pi*i/(4 + a)) + exp(-pi*i/(4 + a))) + i*(-exp(pi*i/(4 - a)) + exp(-pi*i/(4 - a)))/(exp(pi*i/(4 - a)) + exp(-pi*i/(4 - a)))
Parte trigonométrica [src]
     /  pi  \      /  pi \
- tan|------| + tan|-----|
     \-4 + a/      \4 + a/
$$- \tan{\left(\frac{\pi}{a - 4} \right)} + \tan{\left(\frac{\pi}{a + 4} \right)}$$
     2/  pi \        2/  pi \
2*sin |-----|   2*sin |-----|
      \4 + a/         \4 - a/
------------- + -------------
     / 2*pi\         / 2*pi\ 
  sin|-----|      sin|-----| 
     \4 + a/         \4 - a/ 
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4 - a} \right)}}{\sin{\left(\frac{2 \pi}{4 - a} \right)}} + \frac{2 \sin^{2}{\left(\frac{\pi}{a + 4} \right)}}{\sin{\left(\frac{2 \pi}{a + 4} \right)}}$$
   /  pi \      /  pi \
sec|-----|   sec|-----|
   \4 + a/      \4 - a/
---------- + ----------
   /  pi \      /  pi \
csc|-----|   csc|-----|
   \4 + a/      \4 - a/
$$\frac{\sec{\left(\frac{\pi}{a + 4} \right)}}{\csc{\left(\frac{\pi}{a + 4} \right)}} + \frac{\sec{\left(\frac{\pi}{4 - a} \right)}}{\csc{\left(\frac{\pi}{4 - a} \right)}}$$
       /  pi \            /  pi  \    
    sec|-----|         sec|------|    
       \4 + a/            \-4 + a/    
----------------- - ------------------
   /  pi     pi \      /  pi     pi  \
sec|- -- + -----|   sec|- -- + ------|
   \  2    4 + a/      \  2    -4 + a/
$$- \frac{\sec{\left(\frac{\pi}{a - 4} \right)}}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{a - 4} \right)}} + \frac{\sec{\left(\frac{\pi}{a + 4} \right)}}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{a + 4} \right)}}$$
   /  pi     pi \      /  pi     pi \
cos|- -- + -----|   cos|- -- + -----|
   \  2    4 + a/      \  2    4 - a/
----------------- + -----------------
       /  pi \             /  pi \   
    cos|-----|          cos|-----|   
       \4 + a/             \4 - a/   
$$\frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{a + 4} \right)}}{\cos{\left(\frac{\pi}{a + 4} \right)}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{4 - a} \right)}}{\cos{\left(\frac{\pi}{4 - a} \right)}}$$
   /pi     pi \      /pi     pi \
csc|-- - -----|   csc|-- - -----|
   \2    4 + a/      \2    4 - a/
--------------- + ---------------
      /  pi \           /  pi \  
   csc|-----|        csc|-----|  
      \4 + a/           \4 - a/  
$$\frac{\csc{\left(\frac{\pi}{2} - \frac{\pi}{a + 4} \right)}}{\csc{\left(\frac{\pi}{a + 4} \right)}} + \frac{\csc{\left(\frac{\pi}{2} - \frac{\pi}{4 - a} \right)}}{\csc{\left(\frac{\pi}{4 - a} \right)}}$$
       2/  pi  \        2/  pi \
  2*sin |------|   2*sin |-----|
        \-4 + a/         \4 + a/
- -------------- + -------------
      / 2*pi \          / 2*pi\ 
   sin|------|       sin|-----| 
      \-4 + a/          \4 + a/ 
$$- \frac{2 \sin^{2}{\left(\frac{\pi}{a - 4} \right)}}{\sin{\left(\frac{2 \pi}{a - 4} \right)}} + \frac{2 \sin^{2}{\left(\frac{\pi}{a + 4} \right)}}{\sin{\left(\frac{2 \pi}{a + 4} \right)}}$$
    1             1     
---------- - -----------
   /  pi \      /  pi  \
cot|-----|   cot|------|
   \4 + a/      \-4 + a/
$$\frac{1}{\cot{\left(\frac{\pi}{a + 4} \right)}} - \frac{1}{\cot{\left(\frac{\pi}{a - 4} \right)}}$$
    1            1     
---------- + ----------
   /  pi \      /  pi \
cot|-----|   cot|-----|
   \4 + a/      \4 - a/
$$\frac{1}{\cot{\left(\frac{\pi}{a + 4} \right)}} + \frac{1}{\cot{\left(\frac{\pi}{4 - a} \right)}}$$
   /  pi     pi \      /  pi     pi  \
cos|- -- + -----|   cos|- -- + ------|
   \  2    4 + a/      \  2    -4 + a/
----------------- - ------------------
       /  pi \            /  pi  \    
    cos|-----|         cos|------|    
       \4 + a/            \-4 + a/    
$$\frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{a + 4} \right)}}{\cos{\left(\frac{\pi}{a + 4} \right)}} - \frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{a - 4} \right)}}{\cos{\left(\frac{\pi}{a - 4} \right)}}$$
       /  pi \             /  pi \   
    sec|-----|          sec|-----|   
       \4 + a/             \4 - a/   
----------------- + -----------------
   /  pi     pi \      /  pi     pi \
sec|- -- + -----|   sec|- -- + -----|
   \  2    4 + a/      \  2    4 - a/
$$\frac{\sec{\left(\frac{\pi}{4 - a} \right)}}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{4 - a} \right)}} + \frac{\sec{\left(\frac{\pi}{a + 4} \right)}}{\sec{\left(- \frac{\pi}{2} + \frac{\pi}{a + 4} \right)}}$$
   /  pi \      /  pi \
sin|-----|   sin|-----|
   \4 + a/      \4 - a/
---------- + ----------
   /  pi \      /  pi \
cos|-----|   cos|-----|
   \4 + a/      \4 - a/
$$\frac{\sin{\left(\frac{\pi}{4 - a} \right)}}{\cos{\left(\frac{\pi}{4 - a} \right)}} + \frac{\sin{\left(\frac{\pi}{a + 4} \right)}}{\cos{\left(\frac{\pi}{a + 4} \right)}}$$
   /pi     pi \      /pi     pi  \
csc|-- - -----|   csc|-- - ------|
   \2    4 + a/      \2    -4 + a/
--------------- - ----------------
      /  pi \          /  pi  \   
   csc|-----|       csc|------|   
      \4 + a/          \-4 + a/   
$$\frac{\csc{\left(\frac{\pi}{2} - \frac{\pi}{a + 4} \right)}}{\csc{\left(\frac{\pi}{a + 4} \right)}} - \frac{\csc{\left(\frac{\pi}{2} - \frac{\pi}{a - 4} \right)}}{\csc{\left(\frac{\pi}{a - 4} \right)}}$$
csc(pi/2 - pi/(4 + a))/csc(pi/(4 + a)) - csc(pi/2 - pi/(-4 + a))/csc(pi/(-4 + a))