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verificar (cos(3x)-cos(5x))/(sin(3x)+sin(5x))=tan(x)

Solución

verificar

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

Solución

Verdadero

Pasos de solución

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

Manipular el lado derecho

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )}

Re-escribir usando identidades trigonométricas

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )}

Utilizar la identidad suma-producto:

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

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )}

Simplificar

\frac{cos ( 3 x ) - cos ( 5 x )}{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )} : \frac{cos ( 3 x ) - cos ( 5 x )}{2 cos ( x ) sin ( 4 x )}

\frac{cos ( 3 x ) - cos ( 5 x )}{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )}

2 sin (\frac{3 x + 5 x}{2}) cos (\frac{3 x - 5 x}{2}) = 2 sin (\frac{8 x}{2}) cos (- \frac{2 x}{2})

2 sin (\frac{3 x + 5 x}{2}) cos (\frac{3 x - 5 x}{2})

Sumar elementos similares:

3 x + 5 x = 8 x

= 2 sin (\frac{8 x}{2}) cos (\frac{3 x - 5 x}{2})

\frac{3 x - 5 x}{2} = - \frac{2 x}{2}

\frac{3 x - 5 x}{2}

Sumar elementos similares:

3 x - 5 x = - 2 x

= \frac{- 2 x}{2}

Aplicar las propiedades de las fracciones:

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

= - \frac{2 x}{2}

= 2 sin (\frac{8 x}{2}) cos (- \frac{2 x}{2})

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 sin ( \frac{8 x}{2} ) cos ( - \frac{2 x}{2} )}

Utilizar la razón trigonométrica de ángulo negativo:

cos (- x) = cos (x)

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 cos ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}

Dividir:

\frac{2}{2} = 1

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 cos ( x ) sin ( \frac{8 x}{2} )}

Dividir:

\frac{8}{2} = 4

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 cos ( x ) sin ( 4 x )}

= \frac{cos ( 3 x ) - cos ( 5 x )}{2 cos ( x ) sin ( 4 x )}

Utilizar la identidad suma-producto:

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

= \frac{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )}{2 cos ( x ) sin ( 4 x )}

Simplificar

\frac{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )}{2 cos ( x ) sin ( 4 x )} : \frac{sin ( x )}{cos ( x )}

\frac{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )}{2 cos ( x ) sin ( 4 x )}

- 2 sin (\frac{3 x + 5 x}{2}) sin (\frac{3 x - 5 x}{2}) = - 2 sin (\frac{8 x}{2}) sin (- \frac{2 x}{2})

- 2 sin (\frac{3 x + 5 x}{2}) sin (\frac{3 x - 5 x}{2})

Sumar elementos similares:

3 x + 5 x = 8 x

= - 2 sin (\frac{8 x}{2}) sin (\frac{3 x - 5 x}{2})

\frac{3 x - 5 x}{2} = - \frac{2 x}{2}

\frac{3 x - 5 x}{2}

Sumar elementos similares:

3 x - 5 x = - 2 x

= \frac{- 2 x}{2}

Aplicar las propiedades de las fracciones:

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

= - \frac{2 x}{2}

= - 2 sin (\frac{8 x}{2}) sin (- \frac{2 x}{2})

= \frac{- 2 sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}{2 cos ( x ) sin ( 4 x )}

Aplicar las propiedades de las fracciones:

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

= - \frac{2 sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}{2 cos ( x ) sin ( 4 x )}

Dividir:

\frac{2}{2} = 1

= - \frac{sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}{cos ( x ) sin ( 4 x )}

Utilizar la razón trigonométrica de ángulo negativo:

sin (- x) = - sin (x)

= - \frac{( - sin ( \frac{2 x}{2} ) ) sin ( \frac{8 x}{2} )}{cos ( x ) sin ( 4 x )}

Quitar los parentesis:

(- a) = - a

= - \frac{- sin ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}{cos ( x ) sin ( 4 x )}

Aplicar las propiedades de las fracciones:

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

= - (- \frac{sin ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}{cos ( x ) sin ( 4 x )})

Aplicar la regla

- (- a) = a

= \frac{sin ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}{cos ( x ) sin ( 4 x )}

Dividir:

\frac{2}{2} = 1

= \frac{sin ( x ) sin ( \frac{8 x}{2} )}{cos ( x ) sin ( 4 x )}

Dividir:

\frac{8}{2} = 4

= \frac{sin ( x ) sin ( 4 x )}{cos ( x ) sin ( 4 x )}

Eliminar los terminos comunes:

sin (4 x)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= tan (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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