Дано

$$sin{left (a + frac{30 pi}{360} right )} – frac{cos{left (a + frac{60 pi}{360} right )}}{sin{left (a + frac{30 pi}{360} right )}} + cos{left (a + frac{60 pi}{360} right )}$$
Подстановка условия
$$sin{left (a + frac{30 pi}{360} right )} – frac{cos{left (a + frac{60 pi}{360} right )}}{sin{left (a + frac{30 pi}{360} right )}} + cos{left (a + frac{60 pi}{360} right )}$$

sin((30*pi)/360 + (-4)) – cos((60*pi)/360 + (-4))/sin((30*pi)/360 + (-4))^1 + cos((60*pi)/360 + (-4))

$$sin{left ((-4) + frac{30 pi}{360} right )} – frac{cos{left ((-4) + frac{60 pi}{360} right )}}{sin{left ((-4) + frac{30 pi}{360} right )}} + cos{left ((-4) + frac{60 pi}{360} right )}$$

sin((30*pi)/360 – 4) – cos((60*pi)/360 – 4)/sin((30*pi)/360 – 4)^1 + cos((60*pi)/360 – 4)

$$cos{left (-4 + frac{60 pi}{360} right )} + sin{left (-4 + frac{30 pi}{360} right )} – frac{cos{left (-4 + frac{60 pi}{360} right )}}{sin{left (-4 + frac{30 pi}{360} right )}}$$

-sin(4 + pi/3)/cos(4 + 5*pi/12) + cos(4 + 5*pi/12) + sin(4 + pi/3)

$$sin{left (frac{pi}{3} + 4 right )} + cos{left (frac{5 pi}{12} + 4 right )} – frac{sin{left (frac{pi}{3} + 4 right )}}{cos{left (frac{5 pi}{12} + 4 right )}}$$
Степени
$$sin{left (a + frac{pi}{12} right )} + cos{left (a + frac{pi}{6} right )} – frac{cos{left (a + frac{pi}{6} right )}}{sin{left (a + frac{pi}{12} right )}}$$
Численный ответ

-cos((60*pi)/360 + a)/sin((30*pi)/360 + a) + cos((60*pi)/360 + a) + sin((30*pi)/360 + a)

Рациональный знаменатель
$$frac{1}{sin{left (a + frac{pi}{12} right )}} left(sin^{2}{left (a + frac{pi}{12} right )} + sin{left (a + frac{pi}{12} right )} cos{left (a + frac{pi}{6} right )} – cos{left (a + frac{pi}{6} right )}right)$$
Объединение рациональных выражений
$$frac{1}{sin{left (frac{1}{12} left(12 a + piright) right )}} left(sin^{2}{left (frac{1}{12} left(12 a + piright) right )} + sin{left (frac{1}{12} left(12 a + piright) right )} cos{left (frac{1}{6} left(6 a + piright) right )} – cos{left (frac{1}{6} left(6 a + piright) right )}right)$$
Общее упрощение

/60*pi
cos|—– + a|
360 / / pi / pi
– ————– + cos|a + –| + sin|a + –|
/30*pi 6 / 12/
sin|—– + a|
360 /

$$sin{left (a + frac{pi}{12} right )} + cos{left (a + frac{pi}{6} right )} – frac{cos{left (a + frac{60 pi}{360} right )}}{sin{left (a + frac{30 pi}{360} right )}}$$
Соберем выражение
$$sin{left (a + frac{pi}{12} right )} – cos{left (a + frac{pi}{6} right )} csc{left (a + frac{pi}{12} right )} + cos{left (a + frac{pi}{6} right )}$$

/60*pi
cos|—– + a|
360 / /60*pi /30*pi
– ————— + cos|—– + a| + sin|—– + a|
1/30*pi 360 / 360 /
sin |—– + a|
360 /

$$sin{left (a + frac{30 pi}{360} right )} + cos{left (a + frac{60 pi}{360} right )} – frac{cos{left (a + frac{60 pi}{360} right )}}{sin{left (a + frac{30 pi}{360} right )}}$$
Общий знаменатель

/ pi
cos|a + –|
6 / / pi / pi
– ———– + cos|a + –| + sin|a + –|
/ pi 6 / 12/
sin|a + –|
12/

$$sin{left (a + frac{pi}{12} right )} + cos{left (a + frac{pi}{6} right )} – frac{cos{left (a + frac{pi}{6} right )}}{sin{left (a + frac{pi}{12} right )}}$$
Тригонометрическая часть

/ pi
cos|a + –|
6 / / pi / pi
– ———– + cos|a + –| + sin|a + –|
/ pi 6 / 12/
sin|a + –|
12/

$$sin{left (a + frac{pi}{12} right )} + cos{left (a + frac{pi}{6} right )} – frac{cos{left (a + frac{pi}{6} right )}}{sin{left (a + frac{pi}{12} right )}}$$
Комбинаторика

2/ pi / pi / pi / pi
sin |a + –| – cos|a + –| + cos|a + –|*sin|a + –|
12/ 6 / 6 / 12/
—————————————————-
/ pi
sin|a + –|
12/

$$frac{1}{sin{left (a + frac{pi}{12} right )}} left(sin^{2}{left (a + frac{pi}{12} right )} + sin{left (a + frac{pi}{12} right )} cos{left (a + frac{pi}{6} right )} – cos{left (a + frac{pi}{6} right )}right)$$
Раскрыть выражение

___
sin(a) / 3 *cos(a)
—— – ———— / ___ ___ / ___ ___ ___
sin(a) 2 2 | / 2 / 6 | |/ 2 / 6 | / 3 *cos(a)
– —— + ————————————————- + |- —– + —–|*cos(a) + |—– + —–|*sin(a) + ————
2 / ___ ___ / ___ ___ 4 4 / 4 4 / 2
| / 2 / 6 | |/ 2 / 6 |
|- —– + —–|*cos(a) + |—– + —–|*sin(a)
4 4 / 4 4 /

$$- frac{1}{2} sin{left (a right )} + left(frac{sqrt{2}}{4} + frac{sqrt{6}}{4}right) sin{left (a right )} + left(- frac{sqrt{2}}{4} + frac{sqrt{6}}{4}right) cos{left (a right )} + frac{sqrt{3}}{2} cos{left (a right )} + frac{frac{1}{2} sin{left (a right )} – frac{sqrt{3}}{2} cos{left (a right )}}{left(frac{sqrt{2}}{4} + frac{sqrt{6}}{4}right) sin{left (a right )} + left(- frac{sqrt{2}}{4} + frac{sqrt{6}}{4}right) cos{left (a right )}}$$
   
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Llers44
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