1) Вычислим: Sin(p/48) cos(p/48) cos(p/24) cos(p/12)
Заметим, что
Sin(p/24) = 2 Sin(p/48) cos(p/48)
Cos(p/24) = 2 Cos(p/48) cos(p/12)
Подставим:
Sin(p/48) cos(p/48) 2 Cos(p/48) cos(p/12)
= 2 Sin(p/48) cos^2(p/48) cos(p/12)
= Sin(p/24) cos^2(p/24)
= (2 Sin(p/48) cos(p/48)) (2 Cos(p/48) cos(p/12))
= 4 Sin(p/48) cos^2(p/48) cos(p/12)
= 4 Sin(p/24) cos^2(p/24)
= 4 Sin(p/24) (1 - Sin^2(p/24))
= 4 Sin(p/24) - 4 * Sin^3(p/24)

Ответ: 4 Sin(p/24) - 4 Sin^3(p/24)

2) Решим уравнение 26sinxcosx - cos^4(x) + 7 = 0
26sinxcosx = cos^4(x) - 7
26*sin(2x) = cos^4(x) -7
Sin(2x) = (cos^4(x) - 7) / 26
Sin(2x) = (1 - sin^2(x))^2 - 7) / 26

3) Решим уравнение sin^2(2x - pi/6) = 3/4
sin^2(2x - pi/6) = 3/4
sin(2x - pi/6) = sqrt(3) /2
2x - pi/6 = pi/3
2x = pi/3 + pi/6
2x = pi/2
x = pi/4

4) Решим уравнение sin^2(x/2) = cos^2(7x/2)
sin(x/2)^2 = cos(7x/2)^2
1 - cos(x)/2 = sin(7x)/2

разложим cos(7x) в сумму косинусов:
1 - cos(x)/2 = 2sinx(cos^6x - 15sin^4x + 70sin^2(x) -105)

отсюда найдем x.