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Popular Trigonometría >

verificar sec(pi/4+a)sec(pi/4-a)=2sec(2a)

Solución

verificar

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a) = 2 sec (2 a)

Solución

Verdadero

Pasos de solución

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a) = 2 sec (2 a)

Manipular el lado derecho

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a)

Re-escribir usando identidades trigonométricas

sec (\frac{π}{4} - a)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{cos ( \frac{π}{4} - a )}

Utilizar la identidad de diferencia de ángulos:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{1}{cos ( \frac{π}{4} ) cos ( a ) + sin ( \frac{π}{4} ) sin ( a )}

Simplificar

\frac{1}{cos ( \frac{π}{4} ) cos ( a ) + sin ( \frac{π}{4} ) sin ( a )} : \frac{2}{cos ( a ) + sin ( a )}

\frac{1}{cos ( \frac{π}{4} ) cos ( a ) + sin ( \frac{π}{4} ) sin ( a )}

cos (\frac{π}{4}) cos (a) + sin (\frac{π}{4}) sin (a) = \frac{2}{2} cos (a) + \frac{2}{2} sin (a)

cos (\frac{π}{4}) cos (a) + sin (\frac{π}{4}) sin (a)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (a) + sin (\frac{π}{4}) sin (a)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{2}{2}

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (a) + \frac{2}{2} sin (a)

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

Multiplicar

\frac{2}{2} cos (a) : \frac{2 cos ( a )}{2}

\frac{2}{2} cos (a)

Multiplicar fracciones:

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

= \frac{2 cos ( a )}{2}

= \frac{1}{\frac{2 c o s ( a )}{2} + \frac{2}{2} sin ( a )}

Multiplicar

\frac{2}{2} sin (a) : \frac{2 sin ( a )}{2}

\frac{2}{2} sin (a)

Multiplicar fracciones:

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

= \frac{2 sin ( a )}{2}

= \frac{1}{\frac{2 c o s ( a )}{2} + \frac{2 s i n ( a )}{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{2 cos ( a ) + 2 sin ( a )}{2}

Aplicar la regla

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

= \frac{2 cos ( a ) + 2 sin ( a )}{2}

= \frac{1}{\frac{2 c o s ( a ) + 2 s i n ( a )}{2}}

Aplicar las propiedades de las fracciones:

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

= \frac{2}{2 cos ( a ) + 2 sin ( a )}

Factorizar el termino común

2

= \frac{2}{2 ( cos ( a ) + sin ( a ) )}

Cancelar

\frac{2}{2 ( cos ( a ) + sin ( a ) )} : \frac{2}{cos ( a ) + sin ( a )}

\frac{2}{2 ( cos ( a ) + sin ( a ) )}

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

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

Restar:

1 - \frac{1}{2} = \frac{1}{2}

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

Aplicar las leyes de los exponentes:

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

= \frac{2}{cos ( a ) + sin ( a )}

= \frac{2}{cos ( a ) + sin ( a )}

= \frac{2}{cos ( a ) + sin ( a )}

= sec (\frac{π}{4} + a) \frac{2}{cos ( a ) + sin ( a )}

Re-escribir usando identidades trigonométricas

sec (\frac{π}{4} + a)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{cos ( \frac{π}{4} + a )}

Utilizar la identidad de suma de ángulos:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{1}{cos ( \frac{π}{4} ) cos ( a ) - sin ( \frac{π}{4} ) sin ( a )}

Simplificar

\frac{1}{cos ( \frac{π}{4} ) cos ( a ) - sin ( \frac{π}{4} ) sin ( a )} : \frac{2}{cos ( a ) - sin ( a )}

\frac{1}{cos ( \frac{π}{4} ) cos ( a ) - sin ( \frac{π}{4} ) sin ( a )}

cos (\frac{π}{4}) cos (a) - sin (\frac{π}{4}) sin (a) = \frac{2}{2} cos (a) - \frac{2}{2} sin (a)

cos (\frac{π}{4}) cos (a) - sin (\frac{π}{4}) sin (a)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (a) - sin (\frac{π}{4}) sin (a)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{2}{2}

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (a) - \frac{2}{2} sin (a)

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

Multiplicar

\frac{2}{2} cos (a) : \frac{2 cos ( a )}{2}

\frac{2}{2} cos (a)

Multiplicar fracciones:

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

= \frac{2 cos ( a )}{2}

= \frac{1}{\frac{2 c o s ( a )}{2} - \frac{2}{2} sin ( a )}

Multiplicar

\frac{2}{2} sin (a) : \frac{2 sin ( a )}{2}

\frac{2}{2} sin (a)

Multiplicar fracciones:

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

= \frac{2 sin ( a )}{2}

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

Combinar las fracciones usando el mínimo común denominador:

\frac{2 cos ( a ) - 2 sin ( a )}{2}

Aplicar la regla

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

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

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

Aplicar las propiedades de las fracciones:

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

= \frac{2}{2 cos ( a ) - 2 sin ( a )}

Factorizar el termino común

2

= \frac{2}{2 ( cos ( a ) - sin ( a ) )}

Cancelar

\frac{2}{2 ( cos ( a ) - sin ( a ) )} : \frac{2}{cos ( a ) - sin ( a )}

\frac{2}{2 ( cos ( a ) - sin ( a ) )}

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

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

Restar:

1 - \frac{1}{2} = \frac{1}{2}

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

Aplicar las leyes de los exponentes:

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

= \frac{2}{cos ( a ) - sin ( a )}

= \frac{2}{cos ( a ) - sin ( a )}

= \frac{2}{cos ( a ) - sin ( a )}

= \frac{2}{cos ( a ) - sin ( a )} \cdot \frac{2}{cos ( a ) + sin ( a )}

Simplificar

\frac{2}{cos ( a ) - sin ( a )} \cdot \frac{2}{cos ( a ) + sin ( a )} : \frac{2}{( cos ( a ) - sin ( a ) ) ( cos ( a ) + sin ( a ) )}

\frac{2}{cos ( a ) - sin ( a )} \cdot \frac{2}{cos ( a ) + sin ( a )}

Multiplicar fracciones:

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

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

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

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

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

Manipular el lado izquierdo

2 sec (2 a)

Expresar con seno, coseno

2 sec (2 a)

Utilizar la identidad trigonométrica básica:

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

= 2 \cdot \frac{1}{cos ( 2 a )}

Simplificar

2 \cdot \frac{1}{cos ( 2 a )} : \frac{2}{cos ( 2 a )}

2 \cdot \frac{1}{cos ( 2 a )}

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{cos ( 2 a )}

Multiplicar los numeros:

1 \cdot 2 = 2

= \frac{2}{cos ( 2 a )}

= \frac{2}{cos ( 2 a )}

= \frac{2}{cos ( 2 a )}

Re-escribir usando identidades trigonométricas

\frac{2}{cos ( 2 a )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

Factorizar

cos^{2} (a) - sin^{2} (a) : (cos (a) + sin (a)) (cos (a) - sin (a))

cos^{2} (a) - sin^{2} (a)

Aplicar la siguiente regla para binomios al cuadrado:

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

cos^{2} (a) - sin^{2} (a) = (cos (a) + sin (a)) (cos (a) - sin (a))

= (cos (a) + sin (a)) (cos (a) - sin (a))

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

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