1. a) 4tgx-3ctgx=1 б) 5cosx + 2sinx=0 в) г) д)

a) 4tgx - 3ctgx = 1.

Умножим обе части уравнения на tgx:

4tgx - 3ctgx = 1;

4tg^2 (x) - 3 = tgx;

4tg^2 (x) - tgx - 3 = 0;

D = 1 + 4 * 4 * 3 = 49;

√D = 7;

tgx = (1 ± 7) / 8;

1) tgx = (1 - 7) / 8 = - 6/8 = - 3/4;

x = - arctg (3/4) + πk, k ∈ Z.

2) tgx = (1 + 7) / 8 = 1;

x = π/4 + πk, k ∈ Z.

б) 5cosx + 2sinx = 0.

Разделим обе части уравнения на cosx:

5cosx + 2sinx = 0;

5 + 2tgx = 0;

2tgx = - 5;

tgx = - 5/2;

x = - arctg (5/2) + πk, k ∈ Z.

Ответ: a) - arctg (3/4) + πk; π/4 + πk, k ∈ Z. b) - arctg (5/2) + πk, k ∈ Z.