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tan^2(x)cos(x)+cos^2(x)=1

Solución

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

Solución

x = 2 πn, x = π + 2 πn

Grados

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

Pasos de solución

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

Restar

1

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

- 1 + (\frac{1}{sec ( x )})^{2} + \frac{1}{sec ( x )} tan^{2} (x) : - 1 + \frac{1}{sec ^{2} ( x )} + \frac{tan ^{2} ( x )}{sec ( x )}

- 1 + (\frac{1}{sec ( x )})^{2} + \frac{1}{sec ( x )} tan^{2} (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{sec ^{2} ( x )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

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

\frac{1}{sec ( x )} tan^{2} (x)

Multiplicar fracciones:

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

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

Multiplicar:

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

= \frac{tan ^{2} ( x )}{sec ( x )}

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

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

Utilizar la identidad pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

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

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

Usando el método de sustitución

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

Sea:

sec (x) = u

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0 : u = 1, u = - 1

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

Multiplicar por el mínimo común múltiplo

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

Encontrar el mínimo común múltiplo de

u, u^{2} : u^{2}

u, u^{2}

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

u

u^{2}

= u^{2}

Multiplicar por el mínimo común múltiplo=

u^{2}

- 1 \cdot u^{2} + \frac{- 1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplificar

- 1 \cdot u^{2} + \frac{- 1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplificar

- 1 \cdot u^{2} : - u^{2}

- 1 \cdot u^{2}

Multiplicar:

1 \cdot u^{2} = u^{2}

= - u^{2}

Simplificar

\frac{- 1 + u ^{2}}{u} u^{2} : u (u^{2} - 1)

\frac{- 1 + u ^{2}}{u} u^{2}

Multiplicar fracciones:

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

= \frac{( - 1 + u ^{2} ) u ^{2}}{u}

Eliminar los terminos comunes:

u

= u (u^{2} - 1)

Simplificar

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiplicar fracciones:

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

= \frac{1 \cdot u ^{2}}{u ^{2}}

Eliminar los terminos comunes:

u^{2}

= 1

Simplificar

0 \cdot u^{2} : 0

0 \cdot u^{2}

Aplicar la regla

0 \cdot a = 0

= 0

- u^{2} + u (u^{2} - 1) + 1 = 0

- u^{2} + u (u^{2} - 1) + 1 = 0

- u^{2} + u (u^{2} - 1) + 1 = 0

Resolver

- u^{2} + u (u^{2} - 1) + 1 = 0 : u = 1, u = - 1

- u^{2} + u (u^{2} - 1) + 1 = 0

Desarrollar

- u^{2} + u (u^{2} - 1) + 1 : - u^{2} + u^{3} - u + 1

- u^{2} + u (u^{2} - 1) + 1

Expandir

u (u^{2} - 1) : u^{3} - u

u (u^{2} - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = u, b = u^{2}, c = 1

= u u^{2} - u \cdot 1

= u^{2} u - 1 \cdot u

Simplificar

u^{2} u - 1 \cdot u : u^{3} - u

u^{2} u - 1 \cdot u

u^{2} u = u^{3}

u^{2} u

Aplicar las leyes de los exponentes:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Sumar:

2 + 1 = 3

= u^{3}

1 \cdot u = u

1 \cdot u

Multiplicar:

1 \cdot u = u

= u

= u^{3} - u

= u^{3} - u

= - u^{2} + u^{3} - u + 1

- u^{2} + u^{3} - u + 1 = 0

Escribir en la forma binómica

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{3} - u^{2} - u + 1 = 0

Factorizar

u^{3} - u^{2} - u + 1 : (u - 1)^{2} (u + 1)

u^{3} - u^{2} - u + 1

= (u^{3} - u^{2}) + (- u + 1)

Factorizar

- 1

- u + 1

- (u - 1)

- u + 1

Factorizar el termino común

- 1

= - (u - 1)

Factorizar

u^{2}

u^{3} - u^{2}

u^{2} (u - 1)

u^{3} - u^{2}

Aplicar las leyes de los exponentes:

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

u^{3} = u u^{2}

= u u^{2} - u^{2}

Factorizar el termino común

u^{2}

= u^{2} (u - 1)

= - (u - 1) + u^{2} (u - 1)

Factorizar el termino común

u - 1

= (u - 1) (u^{2} - 1)

Factorizar

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Reescribir

1

como

1^{2}

= u^{2} - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

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

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u - 1) (u + 1) (u - 1)

Simplificar

= (u - 1)^{2} (u + 1)

(u - 1)^{2} (u + 1) = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

u - 1 = 0 or u + 1 = 0

Resolver

u - 1 = 0 : u = 1

u - 1 = 0

Mover

1

al lado derecho

u - 1 = 0

Sumar

1

a ambos lados

u - 1 + 1 = 0 + 1

Simplificar

u = 1

u = 1

Resolver

u + 1 = 0 : u = - 1

u + 1 = 0

Mover

1

al lado derecho

u + 1 = 0

Restar

1

de ambos lados

u + 1 - 1 = 0 - 1

Simplificar

u = - 1

u = - 1

Las soluciones son

u = 1, u = - 1

u = 1, u = - 1

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

u = 0

Tomar el(los) denominador(es) de

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}}

y comparar con cero

u = 0

Resolver

u^{2} = 0 : u = 0

u^{2} = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

u = 0

Los siguientes puntos no están definidos

u = 0

Combinar los puntos no definidos con las soluciones:

u = 1, u = - 1

Sustituir en la ecuación

u = sec (x)

sec (x) = 1, sec (x) = - 1

sec (x) = 1, sec (x) = - 1

sec (x) = 1 : x = 2 πn

sec (x) = 1

Soluciones generales para

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

Soluciones generales para

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

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tan(x/2)-csc(x)=0

tan (\frac{x}{2}) - csc (x) = 0

solvefor x,f=e^{sin(x^2)}resolver para

x, f = e^{s i n (x^{2})}