Решите уравнение (sqrt(2))/(sin x) + (sqrt(2))/(cos x) = (1)/(sin^2 x) + (1)/(cos^2 x) .

Задача №17587: Уравнение

Решите уравнение $\frac{2}{sin x} + \frac{2}{cos x} = \frac{1}{sin ^{2} x} + \frac{1}{cos ^{2} x} .$

\frac{2}{sin x} + \frac{2}{cos x} = \frac{1}{sin ^{2} x} + \frac{1}{cos ^{2} x} ⟺ \frac{\frac{1}{2} ( sin x + cos x )}{2 sin x cos x} = \frac{1}{4 sin ^{2} x cos ^{2} x} ⟺ sin (x + π /4) sin 2 x = 1 ⟺ {sin (x + π /4) = 1 sin 2 x = 1 {sin (x + π /4) = - 1 sin 2 x = - 1 ⟺ {2 x = π /2 + 2 k_{1} π x + π /4 = π /2 + 2 k_{2} π {2 x = - π /2 + 2 k_{1} π x + π /4 = - π /2 + 2 k_{2} π, k_{1}, k_{2} \in Z ⟺ {x = π /4 + k_{1} π x = π /4 + 2 k_{2} π {x = - π /4 + k_{1} π x = - 3 π /4 + 2 k_{2} π, k_{1}, k_{2} \in Z ⟺ [x = π /4 + 2 k π, k \in Z x \in \emptyset ⟺ x = π /4 + 2 k π, k \in Z .

Решите уравнение $\frac{2}{sin x} + \frac{2}{cos x} = \frac{1}{sin ^{2} x} + \frac{1}{cos ^{2} x} .$

\frac{2}{sin x} + \frac{2}{cos x} = \frac{1}{sin ^{2} x} + \frac{1}{cos ^{2} x} ⟺ \frac{\frac{1}{2} ( sin x + cos x )}{2 sin x cos x} = \frac{1}{4 sin ^{2} x cos ^{2} x} ⟺ sin (x + π /4) sin 2 x = 1 ⟺ {sin (x + π /4) = 1 sin 2 x = 1 {sin (x + π /4) = - 1 sin 2 x = - 1 ⟺ {2 x = π /2 + 2 k_{1} π x + π /4 = π /2 + 2 k_{2} π {2 x = - π /2 + 2 k_{1} π x + π /4 = - π /2 + 2 k_{2} π, k_{1}, k_{2} \in Z ⟺ {x = π /4 + k_{1} π x = π /4 + 2 k_{2} π {x = - π /4 + k_{1} π x = - 3 π /4 + 2 k_{2} π, k_{1}, k_{2} \in Z ⟺ [x = π /4 + 2 k π, k \in Z x \in \emptyset ⟺ x = π /4 + 2 k π, k \in Z .