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sec(x)+sin^2(x)+cos^2(x)=tan^2(x)

Solución

sec (x) + sin^{2} (x) + cos^{2} (x) = tan^{2} (x)

Solución

x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Grados

x = 18 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Pasos de solución

sec (x) + sin^{2} (x) + cos^{2} (x) = tan^{2} (x)

Restar

tan^{2} (x)

de ambos lados

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

Expresar con seno, coseno

cos^{2} (x) + sec (x) + sin^{2} (x) - tan^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Convertir a fracción:

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

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

Mínimo común múltiplo de

1, cos (x), 1, cos^{2} (x) : cos^{2} (x)

1, cos (x), 1, cos^{2} (x)

Mínimo común múltiplo (MCM)

Mínimo común múltiplo de

1, 1 : 1

1, 1

Mínimo común múltiplo (MCM)

Descomposición en factores primos de

1

Descomposición en factores primos de

1

Multiplicar cada factor el mayor número de veces que ocurra en cualquier

1

1

= 1

Multiplicar los numeros:

1 = 1

= 1

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

= cos^{2} (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 ^{2} ( x )}{1} :

multiplicar el denominador y el numerador por

cos^{2} (x)

\frac{cos ^{2} ( x )}{1} = \frac{cos ^{2} ( x ) cos ^{2} ( x )}{1 \cdot cos ^{2} ( x )} = \frac{cos ^{4} ( x )}{cos ^{2} ( x )}

Para

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

multiplicar el denominador y el numerador por

cos (x)

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

Para

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

multiplicar el denominador y el numerador por

cos^{2} (x)

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

= \frac{cos ^{4} ( x )}{cos ^{2} ( x )} + \frac{cos ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{cos ^{4} ( x ) + cos ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) + cos ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

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

Factorizar

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

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

Factorizar

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

cos (x) + cos^{4} (x)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

cos^{4} (x) = cos (x) cos^{3} (x)

= cos (x) + cos (x) cos^{3} (x)

Factorizar el termino común

cos (x)

= cos (x) (1 + cos^{3} (x))

Factorizar

cos^{3} (x) + 1 : (cos (x) + 1) (cos^{2} (x) - cos (x) + 1)

1 + cos^{3} (x)

Reescribir

1

como

1^{3}

= cos^{3} (x) + 1^{3}

Aplicar la siguiente regla de productos notables (Suma de cubos):

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

cos^{3} (x) + 1^{3} = (cos (x) + 1) (cos^{2} (x) - cos (x) + 1)

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

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

Factorizar

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

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

Factorizar el termino común

sin^{2} (x)

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

Factorizar

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

- 1 + cos^{2} (x)

Reescribir

1

como

1^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

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

= (cos (x) + 1) (cos (x) - 1)

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

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

Factorizar el termino común

(1 + cos (x))

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

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

Resolver cada parte por separado

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

1 + cos (x) = 0 : x = π + 2 πn

1 + cos (x) = 0

Mover

1

al lado derecho

1 + cos (x) = 0

Restar

1

de ambos lados

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

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

cos (x)

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar:

1 + 1 = 2

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

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

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

Expandir

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

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

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = sin^{2} (x), b = - 1, c = cos (x)

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Multiplicar:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

Expandir

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

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

Aplicar la siguiente regla de productos notables

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

Aplicar las reglas de los signos

+ (- a) = - a

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

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

cos (x) cos (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

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

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

Sumar elementos similares:

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

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

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

Utilizar la identidad pitagórica:

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

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

= 2 cos (x) - 1

2 cos (x) - 1 = 0

Mover

1

al lado derecho

2 cos (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

2 cos (x) = 1

2 cos (x) = 1

Dividir ambos lados entre

2

2 cos (x) = 1

Dividir ambos lados entre

2

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

Simplificar

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

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

Soluciones generales para

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

cos (x)

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combinar toda las soluciones

x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Ejemplos populares

4sin^2(x)-3=4,sin(2x)-3=0

4 sin^{2} (x) - 3 = 4, sin (2 x) - 3 = 0

tan^2(θ)+tan(θ)=6,0<= θ<180

tan^{2} (θ) + tan (θ) = 6, 0 \leq θ < 18 0^{\circ}

sin(2x)=5sin(x)

sin (2 x) = 5 sin (x)

2cos^2(θ)-4cos(θ)-1=-5cos(θ)

2 cos^{2} (θ) - 4 cos (θ) - 1 = - 5 cos (θ)

-2sin^2(x)-sin(x)+3=0

- 2 sin^{2} (x) - sin (x) + 3 = 0