sin(90+a)*sin(180-a)*(tan ... 180+a)+tan(270-a))еслиa=3 (упростите выражение)

Вы ввели

sin(90 + a)*sin(180 - a)*(tan(180 + a) + tan(270 - a))

$$\left(\tan{\left(270 - a \right)} + \tan{\left(a + 180 \right)}\right) \sin{\left(180 - a \right)} \sin{\left(a + 90 \right)}$$

Подстановка условия

sin(90 + a)*sin(180 - a)*(tan(180 + a) + tan(270 - a)) при a = 3

подставляем

sin(90 + a)*sin(180 - a)*(tan(180 + a) + tan(270 - a))

$$\left(\tan{\left(270 - a \right)} + \tan{\left(a + 180 \right)}\right) \sin{\left(180 - a \right)} \sin{\left(a + 90 \right)}$$

(-tan(180 + a) + tan(-270 + a))*sin(-180 + a)*sin(90 + a)

$$\left(\tan{\left(a - 270 \right)} - \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

переменные

a = 3

$$a = 3$$

(-tan(180 + (3)) + tan(-270 + (3)))*sin(-180 + (3))*sin(90 + (3))

$$\left(\tan{\left((3) - 270 \right)} - \tan{\left((3) + 180 \right)}\right) \sin{\left((3) - 180 \right)} \sin{\left((3) + 90 \right)}$$

(-tan(180 + 3) + tan(-270 + 3))*sin(-180 + 3)*sin(90 + 3)

$$\left(- \tan{\left(3 + 180 \right)} + \tan{\left(-270 + 3 \right)}\right) \sin{\left(-180 + 3 \right)} \sin{\left(3 + 90 \right)}$$

-(-tan(183) - tan(267))*sin(93)*sin(177)

$$- \left(- \tan{\left(183 \right)} - \tan{\left(267 \right)}\right) \sin{\left(93 \right)} \sin{\left(177 \right)}$$

Степени

-(-tan(-270 + a) + tan(180 + a))*sin(-180 + a)*sin(90 + a)

$$- \left(- \tan{\left(a - 270 \right)} + \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

                                                                 /  /   I*(270 - a)    I*(-270 + a)\     /   I*(180 + a)    I*(-180 - a)\\ 
 /   I*(-180 + a)    I*(180 - a)\ /   I*(-90 - a)    I*(90 + a)\ |I*\- e            + e            /   I*\- e            + e            /| 
-\- e             + e           /*\- e            + e          /*|---------------------------------- + ----------------------------------| 
                                                                 |    I*(-270 + a)    I*(270 - a)          I*(-180 - a)    I*(180 + a)   | 
                                                                 \   e             + e                    e             + e              / 
-------------------------------------------------------------------------------------------------------------------------------------------
                                                                     4                                                                     

$$- \frac{\left(\frac{i \left(- e^{i \left(270 - a\right)} + e^{i \left(a - 270\right)}\right)}{e^{i \left(270 - a\right)} + e^{i \left(a - 270\right)}} + \frac{i \left(e^{i \left(- a - 180\right)} - e^{i \left(a + 180\right)}\right)}{e^{i \left(- a - 180\right)} + e^{i \left(a + 180\right)}}\right) \left(e^{i \left(180 - a\right)} - e^{i \left(a - 180\right)}\right) \left(- e^{i \left(- a - 90\right)} + e^{i \left(a + 90\right)}\right)}{4}$$

Численный ответ

(tan(180 + a) + tan(270 - a))*sin(90 + a)*sin(180 - a)

Рациональный знаменатель

sin(-180 + a)*sin(90 + a)*tan(-270 + a) - sin(-180 + a)*sin(90 + a)*tan(180 + a)

$$\sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)} \tan{\left(a - 270 \right)} - \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)} \tan{\left(a + 180 \right)}$$

(-tan(180 + a) + tan(-270 + a))*sin(-180 + a)*sin(90 + a)

$$\left(\tan{\left(a - 270 \right)} - \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

Объединение рациональных выражений

-(-tan(-270 + a) + tan(180 + a))*sin(-180 + a)*sin(90 + a)

$$- \left(- \tan{\left(a - 270 \right)} + \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

Общее упрощение

(-tan(180 + a) + tan(-270 + a))*sin(-180 + a)*sin(90 + a)

$$\left(\tan{\left(a - 270 \right)} - \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

Собрать выражение

cos(270)*tan(-270 + a)   cos(-90 + 2*a)*tan(180 + a)   cos(270)*tan(180 + a)   cos(-90 + 2*a)*tan(-270 + a)
---------------------- + --------------------------- - --------------------- - ----------------------------
          2                           2                          2                          2              

$$- \frac{1}{2} \cos{\left (2 a - 90 \right )} \tan{\left (a - 270 \right )} + \frac{1}{2} \cos{\left (2 a - 90 \right )} \tan{\left (a + 180 \right )} + \frac{1}{2} \cos{\left (270 \right )} \tan{\left (a - 270 \right )} - \frac{1}{2} \cos{\left (270 \right )} \tan{\left (a + 180 \right )}$$

(tan(180 + a) + tan(270 - a))*sin(90 + a)*sin(180 - a)

$$\left(\tan{\left (- a + 270 \right )} + \tan{\left (a + 180 \right )}\right) \sin{\left (- a + 180 \right )} \sin{\left (a + 90 \right )}$$

Общий знаменатель

sin(-180 + a)*sin(90 + a)*tan(-270 + a) - sin(-180 + a)*sin(90 + a)*tan(180 + a)

$$\sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)} \tan{\left(a - 270 \right)} - \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)} \tan{\left(a + 180 \right)}$$

Тригонометрическая часть

(tan(180 + a) + tan(270 - a))*sin(90 + a)*sin(180 - a)

$$\left(\tan{\left (- a + 270 \right )} + \tan{\left (a + 180 \right )}\right) \sin{\left (- a + 180 \right )} \sin{\left (a + 90 \right )}$$

Комбинаторика

-(-tan(-270 + a) + tan(180 + a))*sin(-180 + a)*sin(90 + a)

$$- \left(- \tan{\left(a - 270 \right)} + \tan{\left(a + 180 \right)}\right) \sin{\left(a - 180 \right)} \sin{\left(a + 90 \right)}$$

Раскрыть выражение

   2                                   2                                 2                                   2                                 2                                 2                                   2                                   2                                                                                                                                                                                                                                                                                                                                                                   
cos (a)*sin(90)*sin(180)*tan(270)   sin (a)*cos(90)*cos(180)*tan(a)   cos (a)*sin(90)*sin(180)*tan(180)   cos (a)*sin(90)*sin(180)*tan(a)   cos (a)*sin(90)*sin(180)*tan(a)   sin (a)*cos(90)*cos(180)*tan(270)   sin (a)*cos(90)*cos(180)*tan(180)   sin (a)*cos(90)*cos(180)*tan(a)   cos(90)*cos(a)*sin(180)*sin(a)*tan(270)   cos(180)*cos(a)*sin(90)*sin(a)*tan(a)   cos(90)*cos(a)*sin(180)*sin(a)*tan(180)   cos(90)*cos(a)*sin(180)*sin(a)*tan(a)   cos(90)*cos(a)*sin(180)*sin(a)*tan(a)   cos(180)*cos(a)*sin(90)*sin(a)*tan(270)   cos(180)*cos(a)*sin(90)*sin(a)*tan(180)   cos(180)*cos(a)*sin(90)*sin(a)*tan(a)
--------------------------------- + ------------------------------- + --------------------------------- + ------------------------------- - ------------------------------- - --------------------------------- - --------------------------------- - ------------------------------- + --------------------------------------- + ------------------------------------- + --------------------------------------- + ------------------------------------- - ------------------------------------- - --------------------------------------- - --------------------------------------- - -------------------------------------
       1 + tan(270)*tan(a)                1 + tan(270)*tan(a)                1 - tan(180)*tan(a)                1 - tan(180)*tan(a)               1 + tan(270)*tan(a)                1 + tan(270)*tan(a)                 1 - tan(180)*tan(a)                1 - tan(180)*tan(a)                   1 + tan(270)*tan(a)                      1 + tan(270)*tan(a)                      1 - tan(180)*tan(a)                      1 - tan(180)*tan(a)                     1 + tan(270)*tan(a)                      1 + tan(270)*tan(a)                       1 - tan(180)*tan(a)                      1 - tan(180)*tan(a)         

$$\frac{\sin^{2}{\left(a \right)} \cos{\left(90 \right)} \cos{\left(180 \right)} \tan{\left(a \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} - \frac{\sin^{2}{\left(a \right)} \cos{\left(90 \right)} \cos{\left(180 \right)} \tan{\left(270 \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(90 \right)} \sin{\left(a \right)} \cos{\left(180 \right)} \cos{\left(a \right)} \tan{\left(a \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} - \frac{\sin{\left(180 \right)} \sin{\left(a \right)} \cos{\left(90 \right)} \cos{\left(a \right)} \tan{\left(a \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} - \frac{\sin{\left(90 \right)} \sin{\left(a \right)} \cos{\left(180 \right)} \cos{\left(a \right)} \tan{\left(270 \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(180 \right)} \sin{\left(a \right)} \cos{\left(90 \right)} \cos{\left(a \right)} \tan{\left(270 \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} - \frac{\sin{\left(90 \right)} \sin{\left(180 \right)} \cos^{2}{\left(a \right)} \tan{\left(a \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(90 \right)} \sin{\left(180 \right)} \cos^{2}{\left(a \right)} \tan{\left(270 \right)}}{\tan{\left(270 \right)} \tan{\left(a \right)} + 1} - \frac{\sin^{2}{\left(a \right)} \cos{\left(90 \right)} \cos{\left(180 \right)} \tan{\left(a \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} - \frac{\sin^{2}{\left(a \right)} \cos{\left(90 \right)} \cos{\left(180 \right)} \tan{\left(180 \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(180 \right)} \sin{\left(a \right)} \cos{\left(90 \right)} \cos{\left(a \right)} \tan{\left(a \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} - \frac{\sin{\left(90 \right)} \sin{\left(a \right)} \cos{\left(180 \right)} \cos{\left(a \right)} \tan{\left(a \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(180 \right)} \sin{\left(a \right)} \cos{\left(90 \right)} \cos{\left(a \right)} \tan{\left(180 \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} - \frac{\sin{\left(90 \right)} \sin{\left(a \right)} \cos{\left(180 \right)} \cos{\left(a \right)} \tan{\left(180 \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(90 \right)} \sin{\left(180 \right)} \cos^{2}{\left(a \right)} \tan{\left(a \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1} + \frac{\sin{\left(90 \right)} \sin{\left(180 \right)} \cos^{2}{\left(a \right)} \tan{\left(180 \right)}}{- \tan{\left(180 \right)} \tan{\left(a \right)} + 1}$$