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Popular Trigonometría >

verificar sec(2θ)=(sec^2(θ)csc^2(θ))/(csc^2(θ)-sec^2(θ))

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

\frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Expresar con seno, coseno

\frac{csc ^{2} ( θ ) sec ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} sec ^{2} ( θ )}{( \frac{1}{s i n ( θ )} ) ^{2} - sec ^{2} ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} ( \frac{1}{c o s ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2} - ( \frac{1}{c o s ( θ )} ) ^{2}}

Simplificar

\frac{( \frac{1}{s i n ( θ )} ) ^{2} ( \frac{1}{c o s ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2} - ( \frac{1}{c o s ( θ )} ) ^{2}} : \frac{1}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

\frac{( \frac{1}{s i n ( θ )} ) ^{2} ( \frac{1}{c o s ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2} - ( \frac{1}{c o s ( θ )} ) ^{2}}

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

(\frac{1}{sin ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

(\frac{1}{cos ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} ( \frac{1}{c o s ( θ )} ) ^{2}}{\frac{1}{s i n ^{2} ( θ )} - \frac{1}{c o s ^{2} ( θ )}}

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

(\frac{1}{sin ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{( \frac{1}{c o s ( θ )} ) ^{2} \frac{1}{s i n ^{2} ( θ )}}{\frac{1}{s i n ^{2} ( θ )} - \frac{1}{c o s ^{2} ( θ )}}

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

(\frac{1}{cos ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

= \frac{\frac{1}{s i n ^{2} ( θ )} \cdot \frac{1}{c o s ^{2} ( θ )}}{\frac{1}{s i n ^{2} ( θ )} - \frac{1}{c o s ^{2} ( θ )}}

Simplificar

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

en una fracción:

\frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

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

Mínimo común múltiplo de

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

sin^{2} (θ), cos^{2} (θ)

Mínimo común múltiplo (MCM)

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

sin^{2} (θ)

cos^{2} (θ)

= sin^{2} (θ) cos^{2} (θ)

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} ( θ )} :

multiplicar el denominador y el numerador por

cos^{2} (θ)

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

Para

\frac{1}{cos ^{2} ( θ )} :

multiplicar el denominador y el numerador por

sin^{2} (θ)

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

= \frac{\frac{1}{s i n ^{2} ( θ )} \cdot \frac{1}{c o s ^{2} ( θ )}}{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{s i n ^{2} ( θ ) c o s ^{2} ( θ )}}

Multiplicar

\frac{1}{sin ^{2} ( θ )} \cdot \frac{1}{cos ^{2} ( θ )} : \frac{1}{sin ^{2} ( θ ) cos ^{2} ( θ )}

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

1 \cdot 1 = 1

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

= \frac{\frac{1}{s i n ^{2} ( θ ) c o s ^{2} ( θ )}}{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{s i n ^{2} ( θ ) c o s ^{2} ( θ )}}

Dividir fracciones:

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

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

Simplificar

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

Eliminar los terminos comunes:

sin^{2} (θ)

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

Eliminar los terminos comunes:

cos^{2} (θ)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

= \frac{1}{cos ( 2 θ )}

= \frac{1}{cos ( 2 θ )}

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{\frac{1}{s e c ( 2 θ )}}

Simplificar

\frac{1}{\frac{1}{s e c ( 2 θ )}}

Aplicar las propiedades de las fracciones:

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

= \frac{sec ( 2 θ )}{1}

Aplicar la regla

\frac{a}{1} = a

= sec (2 θ)

sec (2 θ)

sec (2 θ)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar 1/(1-sin(r))=sec^2(r)+sec(r)tan(r)

verificar 1/(sec(x))=sec(x)-tan(x)sin(x)

verificar sin(a+b)sin(a-b)=sin^2(b)-sin^2(a)

verificar sec(b)+tan(b)=(cos(b))/(1-sin(b))

verificar cot^4(x)+cot^2(x)=csc^4(x)-csc^2(x)