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

Solución

cos^{2} (x) + \frac{1}{2} + sin^{2} (x) + \frac{1}{2} = 2

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Pasos de solución

cos^{2} (x) + \frac{1}{2} + sin^{2} (x) + \frac{1}{2} = 2

Restar

2

de ambos lados

\frac{2 cos ^{2} ( x ) + 1}{2} + \frac{2 sin ^{2} ( x ) + 1}{2} - 2 = 0

Re-escribir usando identidades trigonométricas

- 2 + \frac{1 + 2 sin ^{2} ( x )}{2} + \frac{3 - 2 sin ^{2} ( x )}{2} = 0

Usando el método de sustitución

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combinar toda las soluciones

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin^{2} (x) - 2 cos (x) - 2 = 0

\frac{1}{2} tan^{2} (x) = 1 - sec (x)

sin (2 x) = sin^{2} (x)

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

cot^{2} (θ) - 3 = 0