Trigonometría Ejemplos

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

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

Step 1

Simplifica el lado izquierdo.

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Simplifica

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

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

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Simplifica cada término.

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Reescribe

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

${(\sin (x) + \cos (x))}^{2}$ como

(\sin (x) + \cos (x)) (\sin (x) + \cos (x))

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ .

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

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

Expande

(\sin (x) + \cos (x)) (\sin (x) + \cos (x))

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Aplica la propiedad distributiva.

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

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

Aplica la propiedad distributiva.

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

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

Aplica la propiedad distributiva.

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

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

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

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

Simplifica y combina los términos similares.

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Simplifica cada término.

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Multiplica

\sin (x) \sin (x)

$\sin (x) \sin (x)$ .

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Eleva

\sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

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

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

Eleva

\sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

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

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

Usa la regla de la potencia

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

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

Suma

1

$1$ y

1

$1$ .

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

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

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

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

Multiplica

\cos (x) \cos (x)

$\cos (x) \cos (x)$ .

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Eleva

\cos (x)

$\cos (x)$ a la potencia de

1

$1$ .

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

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

Eleva

\cos (x)

$\cos (x)$ a la potencia de

1

$1$ .

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

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

Usa la regla de la potencia

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

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

Suma

1

$1$ y

1

$1$ .

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

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

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

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

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

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

Reordena los factores de

\sin (x) \cos (x)

$\sin (x) \cos (x)$ .

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

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

Suma

\cos (x) \sin (x)

$\cos (x) \sin (x)$ y

\cos (x) \sin (x)

$\cos (x) \sin (x)$ .

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

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

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

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

Mueve

\cos^{2} (x)

$\cos^{2} (x)$ .

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

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

Aplica la identidad pitagórica.

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

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

Simplifica cada término.

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Reordena

2 \cos (x)

$2 \cos (x)$ y

\sin (x)

$\sin (x)$ .

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

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

Reordena

\sin (x)

$\sin (x)$ y

2

$2$ .

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

$1 - (1 + 2 \cdot \sin (x) \cos (x)) = - \sin (2 x)$

Aplica la razón del ángulo doble sinusoidal.

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

$1 - (1 + \sin (2 x)) = - \sin (2 x)$

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

$1 - (1 + \sin (2 x)) = - \sin (2 x)$

Aplica la propiedad distributiva.

1 - 1 \cdot 1 - \sin (2 x) = - \sin (2 x)

$1 - 1 \cdot 1 - \sin (2 x) = - \sin (2 x)$

Multiplica

- 1

$- 1$ por

1

$1$ .

1 - 1 - \sin (2 x) = - \sin (2 x)

$1 - 1 - \sin (2 x) = - \sin (2 x)$

1 - 1 - \sin (2 x) = - \sin (2 x)

$1 - 1 - \sin (2 x) = - \sin (2 x)$

Simplifica mediante la resta de números.

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Resta

1

$1$ de

1

$1$ .

0 - \sin (2 x) = - \sin (2 x)

$0 - \sin (2 x) = - \sin (2 x)$

Resta

\sin (2 x)

$\sin (2 x)$ de

0

$0$ .

- \sin (2 x) = - \sin (2 x)

$- \sin (2 x) = - \sin (2 x)$

- \sin (2 x) = - \sin (2 x)

$- \sin (2 x) = - \sin (2 x)$

- \sin (2 x) = - \sin (2 x)

$- \sin (2 x) = - \sin (2 x)$

- \sin (2 x) = - \sin (2 x)

$- \sin (2 x) = - \sin (2 x)$

Step 2

Mueve todos los términos que contengan

\sin (2 x)

$\sin (2 x)$ al lado izquierdo de la ecuación.

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Suma

\sin (2 x)

$\sin (2 x)$ a ambos lados de la ecuación.

- \sin (2 x) + \sin (2 x) = 0

$- \sin (2 x) + \sin (2 x) = 0$

Suma

- \sin (2 x)

$- \sin (2 x)$ y

\sin (2 x)

$\sin (2 x)$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 3

Como

0 = 0

$0 = 0$ , la ecuación siempre será verdadera para cualquier valor de

x

$x$ .

Todos los números reales

Step 4

El resultado puede mostrarse de distintas formas.

Todos los números reales

Notación de intervalo:

(- \infty, \infty)

$(- \infty, \infty)$