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sin(x)-2cos(x)cot(x)=csc(x)

Solución

sin (x) - 2 cos (x) cot (x) = csc (x)

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

Pasos de solución

sin (x) - 2 cos (x) cot (x) = csc (x)

Restar

csc (x)

de ambos lados

sin (x) - 2 cos (x) cot (x) - csc (x) = 0

Expresar con seno, coseno

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

Simplificar

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

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

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

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

Sumar

2 cos^{2} (x)

a ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

(2 cos^{2} (x))^{2}

de ambos lados

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

Factorizar

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

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

Resolver cada parte por separado

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

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

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

Verificar las soluciones sustituyendo en la ecuación original

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

sin (2 x) = cos (x - 6 0^{\circ})

tan (18 0^{\circ} - θ) = 33

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

arccsc (2 x) = \frac{π}{6}

2 csc (θ) = 2 cot (θ) + 3 csc (θ)