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Popular Trigonometría >

verificar csc^2(x)sec^2(x)=sec^2(x)+csc^2(x)

Solución

verificar

csc^{2} (x) sec^{2} (x) = sec^{2} (x) + csc^{2} (x)

Solución

Verdadero

Pasos de solución

csc^{2} (x) sec^{2} (x) = sec^{2} (x) + csc^{2} (x)

Manipular el lado izquierdo

sec^{2} (x) + csc^{2} (x)

Expresar con seno, coseno

csc^{2} (x) + sec^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Mínimo común múltiplo de

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

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

Mínimo común múltiplo (MCM)

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

sin^{2} (x)

cos^{2} (x)

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

multiplicar el denominador y el numerador por

cos^{2} (x)

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

Para

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

multiplicar el denominador y el numerador por

sin^{2} (x)

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{cos ^{2} ( x ) ( \frac{1}{c s c ( x )} ) ^{2}}

Utilizar la identidad trigonométrica básica:

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

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2} ( \frac{1}{c s c ( x )} ) ^{2}}

Simplificar

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2} ( \frac{1}{c s c ( x )} ) ^{2}}

(\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}{( \frac{1}{c s c ( x )} ) ^{2} \frac{1}{s e c ^{2} ( x )}}

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

= \frac{1}{\frac{1}{s e c ^{2} ( x )} \cdot \frac{1}{c s c ^{2} ( x )}}

Multiplicar

\frac{1}{sec ^{2} ( x )} \cdot \frac{1}{csc ^{2} ( x )} : \frac{1}{sec ^{2} ( x ) csc ^{2} ( x )}

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

1 \cdot 1 = 1

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

= \frac{1}{\frac{1}{s e c ^{2} ( x ) c s c ^{2} ( x )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Aplicar la regla

\frac{a}{1} = a

= sec^{2} (x) csc^{2} (x)

sec^{2} (x) csc^{2} (x)

sec^{2} (x) csc^{2} (x)

= csc^{2} (x) sec^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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