1) 10 sin 2x +29 sin x-29 cos x =31

Таки решил!
10sin 2x + 29sin x - 29cos x = 31
20sin x*cos x + 29sin x - 29cos x = 31
29(sin x - cos x) = 31 - 20sin x*cos x
29(sin x - cos x) = 10(sin^2 x + cos^2 x) - 20sin x*cos x + 21
29(sin x - cos x) = 10(sin x - cos x)^2 + 21

Замена sin x - cos x = y
10y^2 - 29y + 21 = 0
D = 29^2 - 4*10*21 = 841 - 840 = 1
y1 = sin x - cos x = (29 - 1)/20 = 28/20 = 7/5
y2 = sin x - cos x = (29 + 1)/20 = 30/20 = 3/2

Есть формула
sin x - cos x = 2*(1/2*sin x - 1/2*cos x) =
= 2*(sin x*cos(pi/4) - cos x*sin(pi/4)) = 2*sin(x - pi/4)

Обретаем х
a) sin x - cos x = 2*sin(x - pi/4) = 7/5
sin(x - pi/4) = 7/(52) = 72/10 0,98995 lt; 1
x - pi/4 = (-1)^n*arcsin (72/10) + pi*n
x = pi/4 + (-1)^n*arcsin (72/10) + pi*n

b) sin x - cos x = 2*sin(x - pi/4) = 3/2
sin(x - pi/4) = 3/(22) = 32/4 1,06066 gt; 1
Решений нет
Ответ: x = pi/4 + (-1)^n*arcsin (72/10) + pi*n

2) 6sin^2 x - 3sin x*cos x - cos^2 x = 1 = sin^2 x + cos^2 x
5sin^2 x - 3sin x*cos x - 2cos^2 x = 0
Разделяем все на cos^2 x
5tg^2 x - 3tg x - 2 = 0
Квадратное уравнение условно tg x
(tg x - 1)(5tg x + 2) = 0
a) tg x = 1; x = pi/4 + pi*k
b) tg x = -2/5; x = -arctg(2/5) + pi*k

3) 3tg^2 x - 4cos^2 x = 8
3sin^2 x / cos^2 x - 4cos^2 x - 8 = 0
Умножаем все на cos^2 x
3sin^2 x - 4cos^4 x - 8cos^2 x = 0
3 - 3cos^2 x - 4cos^4 x - 8cos^2 x = 0
Подмена cos^2 x = y, 0 lt;= y lt;= 1 при любом х
4y^2 + 11y - 3 = 0
(y + 3)(4y - 1) = 0
y1 = cos^2 x = -3 lt; 0 - решений нет
y2 = cos^2 x = 1/4
a) cos x = -1/2; x1 = 2pi/3 + 2pi*k; x2 = 4pi/3 + 2pi*k
b) cos x = 1/2; x3 = pi/3 + 2pi*n; x4 = -pi/3 + 2pi*n

4) 5sin 2x - 12(sin x - cos x) + 12 = 0
Решается также, как 1)
10sin x*cos x + 12 = 12(sin x - cos x)
10sin x*cos x - 5 + 17 = 12(sin x - cos x)
-(5sin^2 x + 5cos^2 x - 10sin x*cos x) + 17 = 12(sin x - cos x)
-5(sin x - cos x)^2 + 17 = 12(sin x - cos x)

Подмена sin x - cos x = y.
-5y^2 + 17 = 12y
5y^2 + 12y - 17 = 0
(y - 1)(5y + 17) = 0

y1 = sin x - cos x = 2*sin(x - pi/4) = 1
sin(x - pi/4) = 1/2
x1 - pi/4 = pi/4 + 2pi*n; x1 = pi/2 + 2pi*n
x2 - pi/4 = 3pi/4 + 2pi*n; x2 = pi + 2pi*n

y2 = sin x - cos x = 2*sin(x - pi/4) = -17/5
sin(x - pi/4) = -17/(52) = -172/10 -2,404 lt; -1
Решений нет
Ответ: x1 = pi/2 + 2pi*n; x2 = pi + 2pi*n