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sec^2(x)-1= 1/(cot(x))

Solución

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

Solución

x = \frac{π}{4} + πn

Grados

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

Pasos de solución

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

cot^{2} (x) (sec^{2} (x) - 1)^{2} - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + (- 1 + sec^{2} (x))^{2} cot^{2} (x)

Utilizar la identidad pitagórica:

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

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

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

(tan^{2} (x))^{2} = tan^{4} (x)

(tan^{2} (x))^{2}

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= tan^{4} (x)

= - 1 + tan^{4} (x) cot^{2} (x)

- 1 + cot^{2} (x) tan^{4} (x) = 0

Factorizar

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

- 1 + cot^{2} (x) tan^{4} (x)

Reescribir

- 1 + cot^{2} (x) tan^{4} (x)

como

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

- 1 + cot^{2} (x) tan^{4} (x)

Aplicar las leyes de los exponentes:

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

tan^{4} (x) = (tan^{2} (x))^{2}

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

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

cot^{2} (x) (tan^{2} (x))^{2} = (cot (x) tan^{2} (x))^{2}

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

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

Resolver cada parte por separado

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

tan^{2} (x) cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

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

Re-escribir usando identidades trigonométricas

1 + cot (x) tan^{2} (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{1}{cot ( x )}

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

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot cot (x) = cot (x)

= \frac{cot ( x )}{cot ^{2} ( x )}

Eliminar los terminos comunes:

cot (x)

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

= 1 + \frac{1}{cot ( x )}

1 + \frac{1}{cot ( x )} = 0

Multiplicar ambos lados por

cot (x)

1 + \frac{1}{cot ( x )} = 0

Multiplicar ambos lados por

cot (x)

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

Simplificar

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

Simplificar

1 \cdot cot (x) : cot (x)

1 \cdot cot (x)

Multiplicar:

1 \cdot cot (x) = cot (x)

= cot (x)

Simplificar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cot (x)

= 1

Simplificar

0 \cdot cot (x) : 0

0 \cdot cot (x)

Aplicar la regla

0 \cdot a = 0

= 0

cot (x) + 1 = 0

cot (x) + 1 = 0

cot (x) + 1 = 0

Mover

1

al lado derecho

cot (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

cot (x) = - 1

cot (x) = - 1

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

cot (x) = 0

Tomar el(los) denominador(es) de

1 + \frac{1}{cot ( x )}

y comparar con cero

cot (x) = 0

Los siguientes puntos no están definidos

cot (x) = 0

Combinar los puntos no definidos con las soluciones:

cot (x) = - 1

Soluciones generales para

cot (x) = - 1

cot (x)

tabla de valores periódicos con

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) - 1 = 0 : x = \frac{π}{4} + πn

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{1}{cot ( x )}

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

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot cot (x) = cot (x)

= \frac{cot ( x )}{cot ^{2} ( x )}

Eliminar los terminos comunes:

cot (x)

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

= - 1 + \frac{1}{cot ( x )}

- 1 + \frac{1}{cot ( x )} = 0

Multiplicar ambos lados por

cot (x)

- 1 + \frac{1}{cot ( x )} = 0

Multiplicar ambos lados por

cot (x)

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

Simplificar

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

Simplificar

- 1 \cdot cot (x) : - cot (x)

- 1 \cdot cot (x)

Multiplicar:

1 \cdot cot (x) = cot (x)

= - cot (x)

Simplificar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cot (x)

= 1

Simplificar

0 \cdot cot (x) : 0

0 \cdot cot (x)

Aplicar la regla

0 \cdot a = 0

= 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

Mover

1

al lado derecho

- cot (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

- cot (x) = - 1

- cot (x) = - 1

Dividir ambos lados entre

- 1

- cot (x) = - 1

Dividir ambos lados entre

- 1

\frac{- cot ( x )}{- 1} = \frac{- 1}{- 1}

Simplificar

cot (x) = 1

cot (x) = 1

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

cot (x) = 0

Tomar el(los) denominador(es) de

- 1 + \frac{1}{cot ( x )}

y comparar con cero

cot (x) = 0

Los siguientes puntos no están definidos

cot (x) = 0

Combinar los puntos no definidos con las soluciones:

cot (x) = 1

Soluciones generales para

cot (x) = 1

cot (x)

tabla de valores periódicos con

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

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

Quitar las que no concuerden con la ecuación.

Verificar la solución

\frac{3 π}{4} + πn :

Falso

\frac{3 π}{4} + πn

Sustituir

n = 1

\frac{3 π}{4} + π 1

Multiplicar

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

por

x = \frac{3 π}{4} + π 1

sec^{2} (\frac{3 π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{3 π}{4} + π 1 )}

Simplificar

1 = - 1

\Rightarrow Falso

Verificar la solución

\frac{π}{4} + πn :

Verdadero

\frac{π}{4} + πn

Sustituir

n = 1

\frac{π}{4} + π 1

Multiplicar

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

por

x = \frac{π}{4} + π 1

sec^{2} (\frac{π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{π}{4} + π 1 )}

Simplificar

1 = 1

\Rightarrow Verdadero

x = \frac{π}{4} + πn

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