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(sin(x))/(1-cos(x))=csc(x)

Solución

\frac{sin ( x )}{1 - cos ( x )} = csc (x)

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

Pasos de solución

\frac{sin ( x )}{1 - cos ( x )} = csc (x)

Restar

csc (x)

de ambos lados

\frac{sin ( x )}{1 - cos ( x )} - csc (x) = 0

Simplificar

\frac{sin ( x )}{1 - cos ( x )} - csc (x) : \frac{sin ( x ) - csc ( x ) ( 1 - cos ( x ) )}{1 - cos ( x )}

\frac{sin ( x ) - csc ( x ) ( 1 - cos ( x ) )}{1 - cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (x) - csc (x) (1 - cos (x)) = 0

Expresar con seno, coseno

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

Simplificar

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Restar

cos (x)

de ambos lados

sin^{2} (x) - 1 = - cos (x)

Elevar al cuadrado ambos lados

(sin^{2} (x) - 1)^{2} = (- cos (x))^{2}

Restar

(- cos (x))^{2}

de ambos lados

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

Factorizar

(sin^{2} (x) - 1)^{2} - cos^{2} (x) : (sin^{2} (x) - 1 + cos (x)) (sin^{2} (x) - 1 - cos (x))

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

Resolver cada parte por separado

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

sin^{2} (x) - 1 + cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn

sin^{2} (x) - 1 - cos (x) = 0 : x = π + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

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

Verificar las soluciones sustituyendo en la ecuación original

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

2 sin^{2} (θ) + 13 sin (θ) + 6 = 0

- 3 tan (- x) = 5 tan (x) - 3

tan (a) = 22

sin (2 x) = 3 cos (x)

cos (2 x) = 685