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1/(1-cos(x))+1/(1+cos(x))=2csc(x)

Solución

\frac{1}{1 - cos ( x )} + \frac{1}{1 + cos ( x )} = 2 csc (x)

x = \frac{π}{2} + 2 πn

Pasos de solución

\frac{1}{1 - cos ( x )} + \frac{1}{1 + cos ( x )} = 2 csc (x)

Restar

2 csc (x)

de ambos lados

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

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

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

Expresar con seno, coseno

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

Simplificar

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

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

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

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

Restar

2 sin (x)

de ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

(2 cos^{2} (x) - 2)^{2} - 4 sin^{2} (x) = 0

Factorizar

(2 cos^{2} (x) - 2)^{2} - 4 sin^{2} (x) : 4 (cos^{2} (x) - 1 + sin (x)) (cos^{2} (x) - 1 - sin (x))

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

Resolver cada parte por separado

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

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

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

Combinar toda las soluciones

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

Verificar las soluciones sustituyendo en la ecuación original

x = \frac{π}{2} + 2 πn

\frac{3}{8} = cos (θ)

cos (θ) = 0.75

cos (χ) = 32

x, tan^{3} (x) = f

4.5 8^{2} = 3. 3^{2} + 3. 3^{2} - 2 (3.3) (3.3) cos (x)