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Popular >

verificar sec^2(B)-csc^2(B)=(tan(B)-cot(B))/(sin(B)cos(B))

Solución

verificar

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

Solución

Verdadero

Pasos de solución

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

Manipular el lado izquierdo

\frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

Expresar con seno, coseno

\frac{- cot ( B ) + tan ( B )}{cos ( B ) sin ( B )}

Utilizar la identidad trigonométrica básica:

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

= \frac{- \frac{c o s ( B )}{s i n ( B )} + tan ( B )}{cos ( B ) sin ( B )}

Utilizar la identidad trigonométrica básica:

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

= \frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

Simplificar

\frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )} : \frac{- cos ^{2} ( B ) + sin ^{2} ( B )}{cos ^{2} ( B ) sin ^{2} ( B )}

\frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

Simplificar

- \frac{cos ( B )}{sin ( B )} + \frac{sin ( B )}{cos ( B )}

en una fracción:

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

- \frac{cos ( B )}{sin ( B )} + \frac{sin ( B )}{cos ( B )}

Mínimo común múltiplo de

sin (B), cos (B) : sin (B) cos (B)

sin (B), cos (B)

Mínimo común múltiplo (MCM)

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

sin (B)

cos (B)

= sin (B) cos (B)

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 ( B )}{sin ( B )} :

multiplicar el denominador y el numerador por

cos (B)

\frac{cos ( B )}{sin ( B )} = \frac{cos ( B ) cos ( B )}{sin ( B ) cos ( B )} = \frac{cos ^{2} ( B )}{sin ( B ) cos ( B )}

Para

\frac{sin ( B )}{cos ( B )} :

multiplicar el denominador y el numerador por

sin (B)

\frac{sin ( B )}{cos ( B )} = \frac{sin ( B ) sin ( B )}{cos ( B ) sin ( B )} = \frac{sin ^{2} ( B )}{sin ( B ) cos ( B )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

= \frac{\frac{- c o s ^{2} ( B ) + s i n ^{2} ( B )}{s i n ( B ) c o s ( B )}}{cos ( B ) sin ( B )}

Aplicar las propiedades de las fracciones:

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

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

sin (B) cos (B) cos (B) sin (B) = cos^{2} (B) sin^{2} (B)

sin (B) cos (B) cos (B) sin (B)

Aplicar las leyes de los exponentes:

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

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

= sin (B) cos^{1 + 1} (B) sin (B)

Sumar:

1 + 1 = 2

= sin (B) cos^{2} (B) sin (B)

Aplicar las leyes de los exponentes:

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

sin (B) sin (B) = sin^{1 + 1} (B)

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

Sumar:

1 + 1 = 2

= cos^{2} (B) sin^{2} (B)

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

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{- cos ^{2} ( B ) + ( \frac{1}{c s c ( B )} ) ^{2}}{cos ^{2} ( B ) ( \frac{1}{c s c ( B )} ) ^{2}}

Utilizar la identidad trigonométrica básica:

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

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

Simplificar

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{c s c ( B )} ) ^{2} \frac{1}{s e c ^{2} ( B )}}

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

Multiplicar

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

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

1 \cdot 1 = 1

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

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

Simplificar

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )}

en una fracción:

\frac{- csc ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )}

Mínimo común múltiplo de

sec^{2} (B), csc^{2} (B) : sec^{2} (B) csc^{2} (B)

sec^{2} (B), csc^{2} (B)

Mínimo común múltiplo (MCM)

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

sec^{2} (B)

csc^{2} (B)

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

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}{sec ^{2} ( B )} :

multiplicar el denominador y el numerador por

csc^{2} (B)

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

Para

\frac{1}{csc ^{2} ( B )} :

multiplicar el denominador y el numerador por

sec^{2} (B)

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

= - \frac{csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )} + \frac{sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- csc ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

= \frac{\frac{- c s c ^{2} ( B ) + s e c ^{2} ( B )}{s e c ^{2} ( B ) c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

Dividir fracciones:

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

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B ) \cdot 1}

Simplificar

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

Eliminar los terminos comunes:

sec^{2} (B)

= - \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) csc ^{2} ( B )}{csc ^{2} ( B )}

Eliminar los terminos comunes:

csc^{2} (B)

= - - csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

= sec^{2} (B) - csc^{2} (B)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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