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Popular Trigonometría >

verificar (1-3cos(x)-4cos^2(x))/(sin^2(x))=(1-4cos(x))/(1-cos(x))

Solución

verificar

\frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )} = \frac{1 - 4 cos ( x )}{1 - cos ( x )}

Solución

Verdadero

Pasos de solución

\frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )} = \frac{1 - 4 cos ( x )}{1 - cos ( x )}

Manipular el lado derecho

\frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )}

Re-escribir usando identidades trigonométricas

\frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )}

Utilizar la identidad pitagórica:

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

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

= \frac{1 - 3 cos ( x ) - 4 ( 1 - sin ^{2} ( x ) )}{sin ^{2} ( x )}

Expandir

1 - 3 cos (x) - 4 (1 - sin^{2} (x)) : 4 sin^{2} (x) - 3 cos (x) - 3

1 - 3 cos (x) - 4 (1 - sin^{2} (x))

Expandir

- 4 (1 - sin^{2} (x)) : - 4 + 4 sin^{2} (x)

- 4 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 4, b = 1, c = sin^{2} (x)

= - 4 \cdot 1 - (- 4) sin^{2} (x)

Aplicar las reglas de los signos

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (x)

Multiplicar los numeros:

4 \cdot 1 = 4

= - 4 + 4 sin^{2} (x)

= 1 - 3 cos (x) - 4 + 4 sin^{2} (x)

Simplificar

1 - 3 cos (x) - 4 + 4 sin^{2} (x) : 4 sin^{2} (x) - 3 cos (x) - 3

1 - 3 cos (x) - 4 + 4 sin^{2} (x)

Agrupar términos semejantes

= - 3 cos (x) + 4 sin^{2} (x) + 1 - 4

Sumar/restar lo siguiente:

1 - 4 = - 3

= 4 sin^{2} (x) - 3 cos (x) - 3

= 4 sin^{2} (x) - 3 cos (x) - 3

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

Utilizar la identidad pitagórica:

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

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

= \frac{4 ( 1 - cos ^{2} ( x ) ) - 3 cos ( x ) - 3}{1 - cos ^{2} ( x )}

Factorizar

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

1 - cos^{2} (x)

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

= \frac{4 ( 1 + cos ( x ) ) ( 1 - cos ( x ) ) - 3 cos ( x ) - 3}{( 1 + cos ( x ) ) ( 1 - cos ( x ) )}

Simplificar

\frac{4 ( 1 + cos ( x ) ) ( 1 - cos ( x ) ) - 3 cos ( x ) - 3}{( 1 + cos ( x ) ) ( 1 - cos ( x ) )} : \frac{- 4 cos ( x ) + 1}{1 - cos ( x )}

\frac{4 ( 1 + cos ( x ) ) ( 1 - cos ( x ) ) - 3 cos ( x ) - 3}{( 1 + cos ( x ) ) ( 1 - cos ( x ) )}

Factorizar

4 (1 + cos (x)) (1 - cos (x)) - 3 cos (x) - 3 : (1 + cos (x)) (- 4 cos (x) + 1)

4 (1 + cos (x)) (1 - cos (x)) - 3 cos (x) - 3

Reescribir como

= 4 (1 + cos (x)) (1 - cos (x)) - 3 cos (x) - 3 \cdot 1

Factorizar el termino común

3

= 4 (1 + cos (x)) (1 - cos (x)) - 3 (cos (x) + 1)

Factorizar el termino común

(1 + cos (x))

= (1 + cos (x)) (4 (1 - cos (x)) - 3)

Expandir

4 (- cos (x) + 1) - 3 : - 4 cos (x) + 1

4 (1 - cos (x)) - 3

Expandir

4 (1 - cos (x)) : 4 - 4 cos (x)

4 (1 - cos (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 1, c = cos (x)

= 4 \cdot 1 - 4 cos (x)

Multiplicar los numeros:

4 \cdot 1 = 4

= 4 - 4 cos (x)

= 4 - 4 cos (x) - 3

Simplificar

4 - 4 cos (x) - 3 : - 4 cos (x) + 1

4 - 4 cos (x) - 3

Agrupar términos semejantes

= - 4 cos (x) + 4 - 3

Sumar/restar lo siguiente:

4 - 3 = 1

= - 4 cos (x) + 1

= - 4 cos (x) + 1

= (cos (x) + 1) (- 4 cos (x) + 1)

= \frac{( 1 + cos ( x ) ) ( - 4 cos ( x ) + 1 )}{( 1 + cos ( x ) ) ( 1 - cos ( x ) )}

Eliminar los terminos comunes:

1 + cos (x)

= \frac{- 4 cos ( x ) + 1}{1 - cos ( x )}

= \frac{- 4 cos ( x ) + 1}{1 - cos ( x )}

= \frac{- 4 cos ( x ) + 1}{1 - cos ( x )}

= \frac{1 - 4 cos ( x )}{1 - cos ( x )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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