Trigonometria Exemplos

$\frac{\sin^{3} (x) + \cos^{3} (x)}{1 - 2 \cos^{2} (x)} = \frac{\sec (x) - \sin (x)}{\tan (x) - 1}$

Etapa 1

Subtraia $\frac{\sec (x) - \sin (x)}{\tan (x) - 1}$ dos dois lados da equação.

$\frac{\sin^{3} (x) + \cos^{3} (x)}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1} = 0$

Etapa 2

Simplifique $\frac{\sin^{3} (x) + \cos^{3} (x)}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1}$ .

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Etapa 2.1

Simplifique cada termo.

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Etapa 2.1.1

Simplifique o numerador.

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Etapa 2.1.1.1

Como os dois termos são cubos perfeitos, fatore usando a fórmula da soma de cubos, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ em que $a = \sin (x)$ e $b = \cos (x)$ .

$\frac{(\sin (x) + \cos (x)) (\sin^{2} (x) - \sin (x) \cos (x) + \cos^{2} (x))}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1} = 0$

Etapa 2.1.1.2

Simplifique.

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Etapa 2.1.1.2.1

Reorganize os termos.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + \sin^{2} (x) + \cos^{2} (x))}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1} = 0$

Etapa 2.1.1.2.2

Aplique a identidade trigonométrica fundamental.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1} = 0$

Etapa 2.1.2

Reescreva $\sec (x)$ em termos de senos e cossenos.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\frac{1}{\cos (x)} - \sin (x)}{\tan (x) - 1} = 0$

Etapa 2.1.3

Reescreva $\tan (x)$ em termos de senos e cossenos.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\frac{1}{\cos (x)} - \sin (x)}{\frac{\sin (x)}{\cos (x)} - 1} = 0$

Etapa 2.1.4

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Etapa 2.1.4.1

Multiplique $\frac{\frac{1}{\cos (x)} - \sin (x)}{\frac{\sin (x)}{\cos (x)} - 1}$ por $\frac{\cos (x)}{\cos (x)}$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - (\frac{\cos (x)}{\cos (x)} \cdot \frac{\frac{1}{\cos (x)} - \sin (x)}{\frac{\sin (x)}{\cos (x)} - 1}) = 0$

Etapa 2.1.4.2

Combine.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\cos (x) (\frac{1}{\cos (x)} - \sin (x))}{\cos (x) (\frac{\sin (x)}{\cos (x)} - 1)} = 0$

Etapa 2.1.5

Aplique a propriedade distributiva.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \sin (x))}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \cdot - 1} = 0$

Etapa 2.1.6

Simplifique cancelando.

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Etapa 2.1.6.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 2.1.6.1.1

Cancele o fator comum.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \sin (x))}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \cdot - 1} = 0$

Etapa 2.1.6.1.2

Reescreva a expressão.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 + \cos (x) (- \sin (x))}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \cdot - 1} = 0$

Etapa 2.1.6.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 2.1.6.2.1

Cancele o fator comum.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 + \cos (x) (- \sin (x))}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \cdot - 1} = 0$

Etapa 2.1.6.2.2

Reescreva a expressão.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 + \cos (x) (- \sin (x))}{\sin (x) + \cos (x) \cdot - 1} = 0$

Etapa 2.1.7

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 - \cos (x) \sin (x)}{\sin (x) + \cos (x) \cdot - 1} = 0$

Etapa 2.1.8

Simplifique o denominador.

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Etapa 2.1.8.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 - \cos (x) \sin (x)}{\sin (x) - 1 \cdot \cos (x)} = 0$

Etapa 2.1.8.2

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} - \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)} = 0$

Etapa 2.2

Para escrever $\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)}$ como fração com um denominador comum, multiplique por $\frac{\sin (x) - \cos (x)}{\sin (x) - \cos (x)}$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} \cdot \frac{\sin (x) - \cos (x)}{\sin (x) - \cos (x)} - \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)} = 0$

Etapa 2.3

Para escrever $- \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)}$ como fração com um denominador comum, multiplique por $\frac{1 - 2 \cos^{2} (x)}{1 - 2 \cos^{2} (x)}$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)} \cdot \frac{\sin (x) - \cos (x)}{\sin (x) - \cos (x)} - \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)} \cdot \frac{1 - 2 \cos^{2} (x)}{1 - 2 \cos^{2} (x)} = 0$

Etapa 2.4

Escreva cada expressão com um denominador comum de $(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 2.4.1

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

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1) (\sin (x) - \cos (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} - \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)} \cdot \frac{1 - 2 \cos^{2} (x)}{1 - 2 \cos^{2} (x)} = 0$

Etapa 2.4.2

Multiplique $\frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)}$ por $\frac{1 - 2 \cos^{2} (x)}{1 - 2 \cos^{2} (x)}$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1) (\sin (x) - \cos (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} - \frac{(1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(\sin (x) - \cos (x)) (1 - 2 \cos^{2} (x))} = 0$

Etapa 2.4.3

Reordene os fatores de $(\sin (x) - \cos (x)) (1 - 2 \cos^{2} (x))$ .

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1) (\sin (x) - \cos (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} - \frac{(1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.5

Combine os numeradores em relação ao denominador comum.

$\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6

Simplifique o numerador.

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Etapa 2.6.1

Expanda $(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)$ usando o método FOIL.

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Etapa 2.6.1.1

Aplique a propriedade distributiva.

$\frac{(\sin (x) (- \sin (x) \cos (x) + 1) + \cos (x) (- \sin (x) \cos (x) + 1)) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.1.2

Aplique a propriedade distributiva.

$\frac{(\sin (x) (- \sin (x) \cos (x)) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x) + 1)) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.1.3

Aplique a propriedade distributiva.

$\frac{(\sin (x) (- \sin (x) \cos (x)) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2

Simplifique cada termo.

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Etapa 2.6.2.1

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{(- \sin (x) \sin (x) \cos (x) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.2

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

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Etapa 2.6.2.2.1

Eleve $\sin (x)$ à potência de $1$ .

$\frac{(- (\sin^{1} (x) \sin (x)) \cos (x) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.2.2

Eleve $\sin (x)$ à potência de $1$ .

$\frac{(- (\sin^{1} (x) \sin^{1} (x)) \cos (x) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.2.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{(- {\sin (x)}^{1 + 1} \cos (x) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.2.4

Some $1$ e $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.3

Multiplique $\sin (x)$ por $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) + \cos (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.4

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - \cos (x) \sin (x) \cos (x) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.5

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

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Etapa 2.6.2.5.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - (\cos^{1} (x) \cos (x)) \sin (x) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.5.2

Eleve $\cos (x)$ à potência de $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - (\cos^{1} (x) \cos^{1} (x)) \sin (x) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.5.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - {\cos (x)}^{1 + 1} \sin (x) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.5.4

Some $1$ e $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - \cos^{2} (x) \sin (x) + \cos (x) \cdot 1) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.2.6

Multiplique $\cos (x)$ por $1$ .

$\frac{(- \sin^{2} (x) \cos (x) + \sin (x) - \cos^{2} (x) \sin (x) + \cos (x)) (\sin (x) - \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.3

Expanda $(- \sin^{2} (x) \cos (x) + \sin (x) - \cos^{2} (x) \sin (x) + \cos (x)) (\sin (x) - \cos (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$\frac{- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) \sin (x) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.4

Combine os termos opostos em $- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) \sin (x) + \cos (x) (- \cos (x))$ .

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Etapa 2.6.4.1

Reorganize os fatores nos termos $\sin (x) (- \cos (x))$ e $\cos (x) \sin (x)$ .

$\frac{- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) \sin (x) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.4.2

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

$\frac{- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + 0 + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.4.3

Some $- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x))$ e $0$ .

$\frac{- \sin^{2} (x) \cos (x) \sin (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5

Simplifique cada termo.

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Etapa 2.6.5.1

Multiplique $\sin^{2} (x)$ por $\sin (x)$ somando os expoentes.

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Etapa 2.6.5.1.1

Mova $\sin (x)$ .

$\frac{- (\sin (x) \sin^{2} (x)) \cos (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.1.2

Multiplique $\sin (x)$ por $\sin^{2} (x)$ .

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Etapa 2.6.5.1.2.1

Eleve $\sin (x)$ à potência de $1$ .

$\frac{- (\sin^{1} (x) \sin^{2} (x)) \cos (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.1.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- {\sin (x)}^{1 + 2} \cos (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.1.3

Some $1$ e $2$ .

$\frac{- \sin^{3} (x) \cos (x) - \sin^{2} (x) \cos (x) (- \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2

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

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Etapa 2.6.5.2.1

Multiplique $- 1$ por $- 1$ .

$\frac{- \sin^{3} (x) \cos (x) + 1 \sin^{2} (x) \cos (x) \cos (x) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2.2

Multiplique $\sin^{2} (x)$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos (x) \cos (x) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2.3

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) (\cos^{1} (x) \cos (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2.4

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) (\cos^{1} (x) \cos^{1} (x)) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2.5

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) {\cos (x)}^{1 + 1} + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.2.6

Some $1$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.3

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

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Etapa 2.6.5.3.1

Eleve $\sin (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{1} (x) \sin (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.3.2

Eleve $\sin (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{1} (x) \sin^{1} (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.3.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + {\sin (x)}^{1 + 1} - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.3.4

Some $1$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin (x) \sin (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.4

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

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Etapa 2.6.5.4.1

Eleve $\sin (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) (\sin^{1} (x) \sin (x)) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.4.2

Eleve $\sin (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) (\sin^{1} (x) \sin^{1} (x)) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.4.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) {\sin (x)}^{1 + 1} - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.4.4

Some $1$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) - \cos^{2} (x) \sin (x) (- \cos (x)) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.5

Multiplique $\cos^{2} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 2.6.5.5.1

Mova $\cos (x)$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) - (\cos (x) \cos^{2} (x)) \sin (x) \cdot - 1 + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.5.2

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

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Etapa 2.6.5.5.2.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) - (\cos^{1} (x) \cos^{2} (x)) \sin (x) \cdot - 1 + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.5.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) - {\cos (x)}^{1 + 2} \sin (x) \cdot - 1 + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.5.3

Some $1$ e $2$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) - \cos^{3} (x) \sin (x) \cdot - 1 + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.6

Multiplique $- \cos^{3} (x) \sin (x) \cdot - 1$ .

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Etapa 2.6.5.6.1

Multiplique $- 1$ por $- 1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + 1 \cos^{3} (x) \sin (x) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.6.2

Multiplique $\cos^{3} (x)$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) + \cos (x) (- \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.7

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos (x) \cos (x) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.8

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

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Etapa 2.6.5.8.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - (\cos^{1} (x) \cos (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.8.2

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - (\cos^{1} (x) \cos^{1} (x)) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.8.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - {\cos (x)}^{1 + 1} - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.5.8.4

Some $1$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.6

Combine os termos opostos em $- \sin^{3} (x) \cos (x) + \sin^{2} (x) \cos^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x)$ .

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Etapa 2.6.6.1

Reorganize os fatores nos termos $\sin^{2} (x) \cos^{2} (x)$ e $- \cos^{2} (x) \sin^{2} (x)$ .

$\frac{- \sin^{3} (x) \cos (x) + \cos^{2} (x) \sin^{2} (x) + \sin^{2} (x) - \cos^{2} (x) \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.6.2

Subtraia $\cos^{2} (x) \sin^{2} (x)$ de $\cos^{2} (x) \sin^{2} (x)$ .

$\frac{- \sin^{3} (x) \cos (x) + 0 + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.6.3

Some $- \sin^{3} (x) \cos (x)$ e $0$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - (1 - \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.7

Aplique a propriedade distributiva.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) + (- 1 \cdot 1 - (- \cos (x) \sin (x))) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.8

Multiplique $- 1$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) + (- 1 - (- \cos (x) \sin (x))) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.9

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

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Etapa 2.6.9.1

Multiplique $- 1$ por $- 1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) + (- 1 + 1 (\cos (x) \sin (x))) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.9.2

Multiplique $\cos (x)$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) + (- 1 + \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.10

Expanda $(- 1 + \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))$ usando o método FOIL.

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Etapa 2.6.10.1

Aplique a propriedade distributiva.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 (1 - 2 \cos^{2} (x)) + \cos (x) \sin (x) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.10.2

Aplique a propriedade distributiva.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 \cdot 1 - 1 (- 2 \cos^{2} (x)) + \cos (x) \sin (x) (1 - 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.10.3

Aplique a propriedade distributiva.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 \cdot 1 - 1 (- 2 \cos^{2} (x)) + \cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) (- 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11

Simplifique cada termo.

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Etapa 2.6.11.1

Multiplique $- 1$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 - 1 (- 2 \cos^{2} (x)) + \cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) (- 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.2

Multiplique $- 2$ por $- 1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) (- 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.3

Multiplique $\cos (x)$ por $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) + \cos (x) \sin (x) (- 2 \cos^{2} (x))}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.4

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) - 2 \cos (x) \sin (x) \cos^{2} (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.5

Multiplique $\cos (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 2.6.11.5.1

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

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) - 2 (\cos^{2} (x) \cos (x)) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.5.2

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

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Etapa 2.6.11.5.2.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) - 2 (\cos^{2} (x) \cos^{1} (x)) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.5.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) - 2 {\cos (x)}^{2 + 1} \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.11.5.3

Some $2$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) + \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x) - 2 \cos^{3} (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.12

Subtraia $2 \cos^{3} (x) \sin (x)$ de $\cos^{3} (x) \sin (x)$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) - \cos^{3} (x) \sin (x) - \cos^{2} (x) - 1 + 2 \cos^{2} (x) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.13

Some $- \cos^{2} (x)$ e $2 \cos^{2} (x)$ .

$\frac{- \sin^{3} (x) \cos (x) + \sin^{2} (x) - \cos^{3} (x) \sin (x) - 1 + \cos^{2} (x) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14

Reescreva $- \sin^{3} (x) \cos (x) + \sin^{2} (x) - \cos^{3} (x) \sin (x) - 1 + \cos^{2} (x) + \cos (x) \sin (x)$ em uma forma fatorada.

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Etapa 2.6.14.1

Reorganize os termos.

$\frac{- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) - 1 + \sin^{2} (x) + \cos^{2} (x) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.2

Aplique a identidade trigonométrica fundamental.

$\frac{- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) - 1 + 1 + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.3

Some $- 1$ e $1$ .

$\frac{- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + 0 + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.4

Some $- \sin^{3} (x) \cos (x)$ e $0$ .

$\frac{- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5

Reescreva $- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)$ em uma forma fatorada.

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Etapa 2.6.14.5.1

Fatore $\sin (x) \cos (x)$ de $- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)$ .

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Etapa 2.6.14.5.1.1

Fatore $\sin (x) \cos (x)$ de $- \sin^{3} (x) \cos (x)$ .

$\frac{\sin (x) \cos (x) (- \sin^{2} (x)) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.1.2

Fatore $\sin (x) \cos (x)$ de $- \cos^{3} (x) \sin (x)$ .

$\frac{\sin (x) \cos (x) (- \sin^{2} (x)) + \sin (x) \cos (x) (- \cos^{2} (x)) + \cos (x) \sin (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.1.3

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

$\frac{\sin (x) \cos (x) (- \sin^{2} (x)) + \sin (x) \cos (x) (- \cos^{2} (x)) + \sin (x) \cos (x) (1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.1.4

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

$\frac{\sin (x) \cos (x) (- \sin^{2} (x) - \cos^{2} (x)) + \sin (x) \cos (x) (1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.1.5

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

$\frac{\sin (x) \cos (x) (- \sin^{2} (x) - \cos^{2} (x) + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.2

Fatore $- 1$ de $- \sin^{2} (x)$ .

$\frac{\sin (x) \cos (x) (- (\sin^{2} (x)) - \cos^{2} (x) + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.3

Fatore $- 1$ de $- \cos^{2} (x)$ .

$\frac{\sin (x) \cos (x) (- (\sin^{2} (x)) - (\cos^{2} (x)) + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.4

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

$\frac{\sin (x) \cos (x) (- (\sin^{2} (x) + \cos^{2} (x)) + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.5

Aplique a identidade trigonométrica fundamental.

$\frac{\sin (x) \cos (x) (- 1 \cdot 1 + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.14.5.6

Multiplique $- 1$ por $1$ .

$\frac{\sin (x) \cos (x) (- 1 + 1)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.15

Some $- 1$ e $1$ .

$\frac{\sin (x) \cos (x) \cdot 0}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.16

Combine expoentes.

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Etapa 2.6.16.1

Multiplique $0$ por $\sin (x)$ .

$\frac{0 \cos (x)}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.6.16.2

Multiplique $0$ por $\cos (x)$ .

$\frac{0}{(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))} = 0$

Etapa 2.7

Divida $0$ por $(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))$ .

$0 = 0$

Etapa 3

Como $0 = 0$ , a equação sempre será verdadeira para qualquer valor de $x$ .

Todos os números reais

Etapa 4

O resultado pode ser mostrado de várias formas.

Todos os números reais

Notação de intervalo:

$(- \infty, \infty)$