Trigonometria Exemplos

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

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

Etapa 1

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados,

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

$a^{2} - b^{2} = (a + b) (a - b)$ em que

a = \sin (x) - \cos (x)

$a = \sin (x) - \cos (x)$ e

b = \sin (x) + \cos (x)

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

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

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

Etapa 2

Simplifique os termos.

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Etapa 2.1

Combine os termos opostos em

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

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

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Etapa 2.1.1

Some

- \cos (x)

$- \cos (x)$ e

\cos (x)

$\cos (x)$ .

(\sin (x) + 0 + \sin (x)) (\sin (x) - \cos (x) - (\sin (x) + \cos (x)))

$(\sin (x) + 0 + \sin (x)) (\sin (x) - \cos (x) - (\sin (x) + \cos (x)))$

Etapa 2.1.2

Some

\sin (x)

$\sin (x)$ e

0

$0$ .

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

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

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

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

Etapa 2.2

Some

\sin (x)

$\sin (x)$ e

\sin (x)

$\sin (x)$ .

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

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

Etapa 2.3

Aplique a propriedade distributiva.

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

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

Etapa 2.4

Combine os termos opostos em

\sin (x) - \cos (x) - \sin (x) - \cos (x)

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

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Etapa 2.4.1

Subtraia

\sin (x)

$\sin (x)$ de

\sin (x)

$\sin (x)$ .

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

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

Etapa 2.4.2

Subtraia

\cos (x)

$\cos (x)$ de

0

$0$ .

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

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

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

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

Etapa 2.5

Subtraia

\cos (x)

$\cos (x)$ de

- \cos (x)

$- \cos (x)$ .

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

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

Etapa 2.6

Simplifique a expressão.

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Etapa 2.6.1

Reordene

2 \sin (x)

$2 \sin (x)$ e

- 2 \cos (x)

$- 2 \cos (x)$ .

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

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

Etapa 2.6.2

Adicione parênteses.

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

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

Etapa 2.6.3

Reordene

\cos (x)

$\cos (x)$ e

2 \sin (x)

$2 \sin (x)$ .

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

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

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

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

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

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

Etapa 3

Aplique a fórmula do arco duplo do seno.

- 2 \sin (2 x)

$- 2 \sin (2 x)$