English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometría >

verificar (1-tan(x))/(1+tan(x))=(1-sin(2x))/(cos(2x))

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{1 - tan ( x )}{1 + tan ( x )}

Expresar con seno, coseno

\frac{1 - tan ( x )}{1 + tan ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{1 - \frac{s i n ( x )}{c o s ( x )}}{1 + \frac{s i n ( x )}{c o s ( x )}}

Simplificar

\frac{1 - \frac{s i n ( x )}{c o s ( x )}}{1 + \frac{s i n ( x )}{c o s ( x )}} : \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

\frac{1 - \frac{s i n ( x )}{c o s ( x )}}{1 + \frac{s i n ( x )}{c o s ( x )}}

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

1 \cdot cos (x) = cos (x)

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

= \frac{1 - \frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x ) + s i n ( x )}{c o s ( x )}}

Simplificar

1 - \frac{sin ( x )}{cos ( x )}

en una fracción:

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

1 - \frac{sin ( x )}{cos ( x )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

1 \cdot cos (x) = cos (x)

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

= \frac{\frac{c o s ( x ) - s i n ( x )}{c o s ( x )}}{\frac{c o s ( x ) + s i n ( x )}{c o s ( x )}}

Dividir fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

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

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

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

Multiplicar

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

por

Eq u a t i o n 1

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

Expandir

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

(cos (x) - sin (x)) (cos (x) - sin (x)) cos (2 x)

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

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

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

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

Aplicar la formula del binomio al cuadrado:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos (x), b = sin (x)

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

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

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

Aplicar la siguiente regla de productos notables

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

Aplicar las reglas de los signos

+ (- a) = - a

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Factorizar

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

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

Factorizar el termino común

cos (2 x)

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

Simplificar

= cos (2 x) (- 2 cos (x) sin (x) + 1)

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

Eliminar los terminos comunes:

cos (2 x)

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

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

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

Expandir

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

(cos (x) + sin (x)) (cos (x) - sin (x))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar csch^2(x)=coth^2(x)-1

verificar (tan(y))/(csc(y))= 1/(cos(y))-1/(sec(y))

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

verificar 1/(sin(x)cot(x))= 1/(cos(x))

verificar (csc^2(x)-1)/(csc^2(x))=cos^2(x)