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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(pi/(2+a))*(-sin(a+3*pi/10))*(-sin(a+pi/5))/cot(a)

¿Cómo vas a descomponer esta cot(pi/(2+a))*(-sin(a+3*pi/10))*(-sin(a+pi/5))/cot(a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /  pi \ /    /    3*pi\\ /    /    pi\\
cot|-----|*|-sin|a + ----||*|-sin|a + --||
   \2 + a/ \    \     10 // \    \    5 //
------------------------------------------
                  cot(a)
−sin⁡(a+3π10)cot⁡(πa+2)(−sin⁡(a+π5))cot⁡(a)\frac{- \sin{\left(a + \frac{3 \pi}{10} \right)} \cot{\left(\frac{\pi}{a + 2} \right)} \left(- \sin{\left(a + \frac{\pi}{5} \right)}\right)}{\cot{\left(a \right)}} 
((cot(pi/(2 + a))*(-sin(a + (3*pi)/10)))*(-sin(a + pi/5)))/cot(a)
Simplificación general [src] 
/   ______________      _______________             \       
|  /          ___      /           ___              |       
\\/  10 - 2*\/ 5   + \/  50 - 10*\/ 5   + 8*sin(2*a)/*tan(a)
------------------------------------------------------------
                             /  pi \                        
                       16*tan|-----|                        
                             \2 + a/
(8sin⁡(2a)+10−25+50−105)tan⁡(a)16tan⁡(πa+2)\frac{\left(8 \sin{\left(2 a \right)} + \sqrt{10 - 2 \sqrt{5}} + \sqrt{50 - 10 \sqrt{5}}\right) \tan{\left(a \right)}}{16 \tan{\left(\frac{\pi}{a + 2} \right)}} 
(sqrt(10 - 2*sqrt(5)) + sqrt(50 - 10*sqrt(5)) + 8*sin(2*a))*tan(a)/(16*tan(pi/(2 + a)))
Unión de expresiones racionales [src] 
   /  pi \    /pi + 5*a\    /3*pi + 10*a\
cot|-----|*sin|--------|*sin|-----------|
   \2 + a/    \   5    /    \     10    /
-----------------------------------------
                  cot(a)
sin⁡(5a+π5)sin⁡(10a+3π10)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(\frac{5 a + \pi}{5} \right)} \sin{\left(\frac{10 a + 3 \pi}{10} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
cot(pi/(2 + a))*sin((pi + 5*a)/5)*sin((3*pi + 10*a)/10)/cot(a)
Denominador racional [src] 
   /  pi \    /    3*pi\    /    pi\
cot|-----|*sin|a + ----|*sin|a + --|
   \2 + a/    \     10 /    \    5 /
------------------------------------
               cot(a)
sin⁡(a+3π10)sin⁡(a+π5)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(a + \frac{3 \pi}{10} \right)} \sin{\left(a + \frac{\pi}{5} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
cot(pi/(2 + a))*sin(a + 3*pi/10)*sin(a + pi/5)/cot(a)
Potencias [src] 
   /  pi \    /    pi\    /    3*pi\
cot|-----|*sin|a + --|*sin|a + ----|
   \2 + a/    \    5 /    \     10 /
------------------------------------
               cot(a)
sin⁡(a+π5)sin⁡(a+3π10)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(a + \frac{\pi}{5} \right)} \sin{\left(a + \frac{3 \pi}{10} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
 /     /     3*pi\      /    3*pi\\ /     /     pi\      /    pi\\            
 |   I*|-a - ----|    I*|a + ----|| |   I*|-a - --|    I*|a + --||            
 |     \      10 /      \     10 /| |     \     5 /      \    5 /|    /  pi \ 
-\- e              + e            /*\- e            + e          /*cot|-----| 
                                                                      \2 + a/ 
------------------------------------------------------------------------------
                                   4*cot(a)
−(−ei(−a−3π10)+ei(a+3π10))(−ei(−a−π5)+ei(a+π5))cot⁡(πa+2)4cot⁡(a)- \frac{\left(- e^{i \left(- a - \frac{3 \pi}{10}\right)} + e^{i \left(a + \frac{3 \pi}{10}\right)}\right) \left(- e^{i \left(- a - \frac{\pi}{5}\right)} + e^{i \left(a + \frac{\pi}{5}\right)}\right) \cot{\left(\frac{\pi}{a + 2} \right)}}{4 \cot{\left(a \right)}} 
-(-exp(i*(-a - 3*pi/10)) + exp(i*(a + 3*pi/10)))*(-exp(i*(-a - pi/5)) + exp(i*(a + pi/5)))*cot(pi/(2 + a))/(4*cot(a))
Combinatoria [src] 
   /  pi \    /    pi\    /    3*pi\
cot|-----|*sin|a + --|*sin|a + ----|
   \2 + a/    \    5 /    \     10 /
------------------------------------
               cot(a)
sin⁡(a+π5)sin⁡(a+3π10)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(a + \frac{\pi}{5} \right)} \sin{\left(a + \frac{3 \pi}{10} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
cot(pi/(2 + a))*sin(a + pi/5)*sin(a + 3*pi/10)/cot(a)
Denominador común [src] 
   /  pi \    /    pi\    /    3*pi\
cot|-----|*sin|a + --|*sin|a + ----|
   \2 + a/    \    5 /    \     10 /
------------------------------------
               cot(a)
sin⁡(a+π5)sin⁡(a+3π10)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(a + \frac{\pi}{5} \right)} \sin{\left(a + \frac{3 \pi}{10} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
cot(pi/(2 + a))*sin(a + pi/5)*sin(a + 3*pi/10)/cot(a)
Abrimos la expresión [src] 
                                ___________                           ___________                                 ___________                                 ___________                   
                               /       ___                           /       ___                                 /       ___                                 /       ___                    
          /  pi \             /  5   \/ 5      2       /  pi \      /  5   \/ 5      2       /  pi \     ___    /  5   \/ 5      2       /  pi \     ___    /  5   \/ 5      2       /  pi \
cos(a)*cot|-----|*sin(a)     /   - - ----- *cos (a)*cot|-----|     /   - - ----- *sin (a)*cot|-----|   \/ 5 *  /   - - ----- *cos (a)*cot|-----|   \/ 5 *  /   - - ----- *sin (a)*cot|-----|
          \2 + a/          \/    8     8               \2 + a/   \/    8     8               \2 + a/         \/    8     8               \2 + a/         \/    8     8               \2 + a/
------------------------ + ----------------------------------- + ----------------------------------- + ----------------------------------------- + -----------------------------------------
         cot(a)                          4*cot(a)                              4*cot(a)                                 4*cot(a)                                    4*cot(a)
58−58sin⁡2(a)cot⁡(πa+2)4cot⁡(a)+558−58sin⁡2(a)cot⁡(πa+2)4cot⁡(a)+sin⁡(a)cos⁡(a)cot⁡(πa+2)cot⁡(a)+58−58cos⁡2(a)cot⁡(πa+2)4cot⁡(a)+558−58cos⁡2(a)cot⁡(πa+2)4cot⁡(a)\frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sin^{2}{\left(a \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{4 \cot{\left(a \right)}} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sin^{2}{\left(a \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{4 \cot{\left(a \right)}} + \frac{\sin{\left(a \right)} \cos{\left(a \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} + \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \cos^{2}{\left(a \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{4 \cot{\left(a \right)}} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \cos^{2}{\left(a \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{4 \cot{\left(a \right)}} 
cos(a)*cot(pi/(2 + a))*sin(a)/cot(a) + sqrt(5/8 - sqrt(5)/8)*cos(a)^2*cot(pi/(2 + a))/(4*cot(a)) + sqrt(5/8 - sqrt(5)/8)*sin(a)^2*cot(pi/(2 + a))/(4*cot(a)) + sqrt(5)*sqrt(5/8 - sqrt(5)/8)*cos(a)^2*cot(pi/(2 + a))/(4*cot(a)) + sqrt(5)*sqrt(5/8 - sqrt(5)/8)*sin(a)^2*cot(pi/(2 + a))/(4*cot(a))
Respuesta numérica [src] 
cot(pi/(2 + a))*sin(a + pi/5)*sin(a + (3*pi)/10)/cot(a)
cot(pi/(2 + a))*sin(a + pi/5)*sin(a + (3*pi)/10)/cot(a)
Compilar la expresión [src] 
   /  pi \    /    pi\    /    3*pi\
cot|-----|*sin|a + --|*sin|a + ----|
   \2 + a/    \    5 /    \     10 /
------------------------------------
               cot(a)
sin⁡(a+3π10)sin⁡(a+π5)cot⁡(πa+2)cot⁡(a)\frac{\sin{\left(a + \frac{3 \pi}{10} \right)} \sin{\left(a + \frac{\pi}{5} \right)} \cot{\left(\frac{\pi}{a + 2} \right)}}{\cot{\left(a \right)}} 
cot(pi/(2 + a))*sin(a + pi/5)*sin(a + (3*pi)/10)/cot(a)
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