Question

4 в степени cos2x + 4 в степени cos^2 x =3

Answer 1

4^(cos(2x)) + 4^(cos^2(x)) = 3
cos(2x) = 2cos^2(x) - 1
0.25*4^(2cos^2(x)) + 4^(cos^2(x)) - 3 = 0
Замена: 4^(cos^2(x)) = t > 1
0.25*t^2 + t - 3 = 0
t^2 + 4t - 12 = 0, D=64,
t1 = -6 < 0 - посторонний корень
t2 = 2
4^(cos^2(x)) = 2
2^(2cos^2(x)) = 2
2cos^2(x) = 1, cos^2(x) = 1/2
cosx = √2/2, x = +-π/4 + 2πk
cosx = -√2/2, x = +-3π/4 + 2πk
Объединяем: x = +-π/4 + πk/2
Ответ: x = +-π/4 + πk/2