cot^2(x)csc^2(x)-cot^2(x)=9

Solución

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Pasos de solución

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

Restar

9

de ambos lados

cot^{2} (x) csc^{2} (x) - cot^{2} (x) - 9 = 0

Re-escribir usando identidades trigonométricas

- 8 + csc^{4} (x) - 2 csc^{2} (x) = 0

Usando el método de sustitución

csc (x) = 2, csc (x) = - 2, csc (x) = 2 i, csc (x) = - 2 i

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = 2 i :

Sin solución

csc (x) = - 2 i :

Sin solución

Combinar toda las soluciones

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (θ) + sin (2 θ) = 0

\frac{π}{4} = arctan (x)

2 cos (x) = cos (2 x)

3 cos^{2} (θ) + cos (θ) = 0

cos^{2} (θ) = 0.75