(sin(45+a)-cos(45+a))/(si ... (45+a)+cos(45+a))еслиa=-4 (упростите выражение)

Вы ввели

sin(45 + a) - cos(45 + a)
-------------------------
sin(45 + a) + cos(45 + a)

$$\frac{\sin{\left(a + 45 \right)} - \cos{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}$$

Подстановка условия

(sin(45 + a) - cos(45 + a))/(sin(45 + a) + cos(45 + a)) при a = -4

подставляем

sin(45 + a) - cos(45 + a)
-------------------------
sin(45 + a) + cos(45 + a)

$$\frac{\sin{\left(a + 45 \right)} - \cos{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}$$

      -1        
----------------
   /         pi\
tan|45 + a + --|
   \         4 /

$$- \frac{1}{\tan{\left(a + \frac{\pi}{4} + 45 \right)}}$$

переменные

a = -4

$$a = -4$$

        -1         
-------------------
   /            pi\
tan|45 + (-4) + --|
   \            4 /

$$- \frac{1}{\tan{\left((-4) + \frac{\pi}{4} + 45 \right)}}$$

      -1        
----------------
   /         pi\
tan|45 - 4 + --|
   \         4 /

$$- \frac{1}{\tan{\left(-4 + \frac{\pi}{4} + 45 \right)}}$$

    -1      
------------
   /     pi\
tan|41 + --|
   \     4 /

$$- \frac{1}{\tan{\left(\frac{\pi}{4} + 41 \right)}}$$

Степени

   I*(-45 - a)    I*(45 + a)     /   I*(-45 - a)    I*(45 + a)\
  e              e             I*\- e            + e          /
- ------------ - ----------- - --------------------------------
       2              2                       2                
---------------------------------------------------------------
  I*(-45 - a)    I*(45 + a)     /   I*(-45 - a)    I*(45 + a)\ 
 e              e             I*\- e            + e          / 
 ------------ + ----------- - -------------------------------- 
      2              2                       2                 

$$\frac{- \frac{i \left(- e^{i \left(- a - 45\right)} + e^{i \left(a + 45\right)}\right)}{2} - \frac{e^{i \left(- a - 45\right)}}{2} - \frac{e^{i \left(a + 45\right)}}{2}}{- \frac{i \left(- e^{i \left(- a - 45\right)} + e^{i \left(a + 45\right)}\right)}{2} + \frac{e^{i \left(- a - 45\right)}}{2} + \frac{e^{i \left(a + 45\right)}}{2}}$$

Численный ответ

(-cos(45 + a) + sin(45 + a))/(cos(45 + a) + sin(45 + a))

Рациональный знаменатель

       sin(45 + a)                 cos(45 + a)       
------------------------- - -------------------------
cos(45 + a) + sin(45 + a)   cos(45 + a) + sin(45 + a)

$$\frac{\sin{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}} - \frac{\cos{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}$$

Общее упрощение

      -1        
----------------
   /         pi\
tan|45 + a + --|
   \         4 /

$$- \frac{1}{\tan{\left(a + \frac{\pi}{4} + 45 \right)}}$$

Общий знаменатель

           2*sin(45 + a)      
-1 + -------------------------
     cos(45 + a) + sin(45 + a)

$$-1 + \frac{2 \sin{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}$$

Тригонометрическая часть

      -1        
----------------
   /         pi\
tan|45 + a + --|
   \         4 /

$$- \frac{1}{\tan{\left (a + \frac{\pi}{4} + 45 \right )}}$$

Комбинаторика

-(-sin(45 + a) + cos(45 + a)) 
------------------------------
  cos(45 + a) + sin(45 + a)   

$$- \frac{- \sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}{\sin{\left(a + 45 \right)} + \cos{\left(a + 45 \right)}}$$

Раскрыть выражение

                          cos(45)*sin(a)                                                      cos(a)*sin(45)                                                      sin(45)*sin(a)                                                      cos(45)*cos(a)                         
----------------------------------------------------------------- + ----------------------------------------------------------------- + ----------------------------------------------------------------- - -----------------------------------------------------------------
cos(45)*cos(a) + cos(45)*sin(a) + cos(a)*sin(45) - sin(45)*sin(a)   cos(45)*cos(a) + cos(45)*sin(a) + cos(a)*sin(45) - sin(45)*sin(a)   cos(45)*cos(a) + cos(45)*sin(a) + cos(a)*sin(45) - sin(45)*sin(a)   cos(45)*cos(a) + cos(45)*sin(a) + cos(a)*sin(45) - sin(45)*sin(a)

$$\frac{\sin{\left(a \right)} \cos{\left(45 \right)}}{- \sin{\left(45 \right)} \sin{\left(a \right)} + \sin{\left(a \right)} \cos{\left(45 \right)} + \cos{\left(45 \right)} \cos{\left(a \right)} + \sin{\left(45 \right)} \cos{\left(a \right)}} + \frac{\sin{\left(45 \right)} \sin{\left(a \right)}}{- \sin{\left(45 \right)} \sin{\left(a \right)} + \sin{\left(a \right)} \cos{\left(45 \right)} + \cos{\left(45 \right)} \cos{\left(a \right)} + \sin{\left(45 \right)} \cos{\left(a \right)}} - \frac{\cos{\left(45 \right)} \cos{\left(a \right)}}{- \sin{\left(45 \right)} \sin{\left(a \right)} + \sin{\left(a \right)} \cos{\left(45 \right)} + \cos{\left(45 \right)} \cos{\left(a \right)} + \sin{\left(45 \right)} \cos{\left(a \right)}} + \frac{\sin{\left(45 \right)} \cos{\left(a \right)}}{- \sin{\left(45 \right)} \sin{\left(a \right)} + \sin{\left(a \right)} \cos{\left(45 \right)} + \cos{\left(45 \right)} \cos{\left(a \right)} + \sin{\left(45 \right)} \cos{\left(a \right)}}$$