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(cos(x))/(csc(x))+(sin(x))/(sec(x))=1

Solución

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

Solución

x = \frac{π}{4} + πn

Grados

x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

1

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

1 = \frac{1}{1}

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

Mínimo común múltiplo de

csc (x), sec (x), 1 : csc (x) sec (x)

csc (x), sec (x), 1

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan en al menos una de las expresiones factorizadas

= csc (x) sec (x)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{cos ( x )}{csc ( x )} :

multiplicar el denominador y el numerador por

sec (x)

\frac{cos ( x )}{csc ( x )} = \frac{cos ( x ) sec ( x )}{csc ( x ) sec ( x )}

Para

\frac{sin ( x )}{sec ( x )} :

multiplicar el denominador y el numerador por

csc (x)

\frac{sin ( x )}{sec ( x )} = \frac{sin ( x ) csc ( x )}{sec ( x ) csc ( x )}

Para

\frac{1}{1} :

multiplicar el denominador y el numerador por

csc (x) sec (x)

\frac{1}{1} = \frac{1 \cdot csc ( x ) sec ( x )}{1 \cdot csc ( x ) sec ( x )} = \frac{csc ( x ) sec ( x )}{csc ( x ) sec ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

cos (x) sec (x) + sin (x) csc (x) - csc (x) sec (x) = 0

Expresar con seno, coseno

cos (x) sec (x) - csc (x) sec (x) + csc (x) sin (x)

Utilizar la identidad trigonométrica básica:

sec (x) = \frac{1}{cos ( x )}

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

cos (x) \frac{1}{cos ( x )} = 1

cos (x) \frac{1}{cos ( x )}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( x )}{cos ( x )}

Eliminar los terminos comunes:

cos (x)

= 1

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x ) cos ( x )}

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

Multiplicar fracciones:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot 1}{sin ( x ) cos ( x )}

Multiplicar los numeros:

1 \cdot 1 = 1

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

\frac{1}{sin ( x )} sin (x) = 1

\frac{1}{sin ( x )} sin (x)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot sin ( x )}{sin ( x )}

Eliminar los terminos comunes:

sin (x)

= 1

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

Agrupar términos semejantes

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

Sumar:

1 + 1 = 2

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

Convertir a fracción:

2 = \frac{2 sin ( x ) cos ( x )}{sin ( x ) cos ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

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

- 1 + 2 cos (x) sin (x) = 0

Re-escribir usando identidades trigonométricas

- 1 + 2 cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

Mover

1

al lado derecho

- 1 + sin (2 x) = 0

Sumar

1

a ambos lados

- 1 + sin (2 x) + 1 = 0 + 1

Simplificar

sin (2 x) = 1

sin (2 x) = 1

Soluciones generales para

sin (2 x) = 1

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

Resolver

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{π}{2} + 2 πn

Dividir ambos lados entre

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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