Trigonometria Exemplos

$\frac{\sin^{2} (x) - \tan (x)}{\cos^{2} (x) - \cot (x)} = \tan^{2} (x)$

Etapa 1

Comece do lado esquerdo.

$\frac{\sin^{2} (x) - \tan (x)}{\cos^{2} (x) - \cot (x)}$

Etapa 2

Converta em senos e cossenos.

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Etapa 2.1

Escreva $\tan (x)$ nos senos e cossenos usando a fórmula do quociente.

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

Etapa 2.2

Escreva $\cot (x)$ nos senos e cossenos usando a fórmula do quociente.

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

Etapa 3

Simplifique.

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Etapa 3.1

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

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Etapa 3.1.1

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

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

Etapa 3.1.2

Combine.

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

Etapa 3.2

Aplique a propriedade distributiva.

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

Etapa 3.3

Simplifique cancelando.

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Etapa 3.3.1

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

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Etapa 3.3.1.1

Mova o negativo de maior ordem em $- \frac{\sin (x)}{\cos (x)}$ para o numerador.

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

Etapa 3.3.1.2

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

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

Etapa 3.3.1.3

Cancele o fator comum.

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

Etapa 3.3.1.4

Reescreva a expressão.

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

Etapa 3.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 3.3.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 3.3.4

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

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

Etapa 3.3.5

Some $1$ e $1$ .

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

Etapa 3.3.6

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

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Etapa 3.3.6.1

Mova o negativo de maior ordem em $- \frac{\cos (x)}{\sin (x)}$ para o numerador.

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

Etapa 3.3.6.2

Cancele o fator comum.

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

Etapa 3.3.6.3

Reescreva a expressão.

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

Etapa 3.3.7

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.3.8

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.3.9

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

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

Etapa 3.3.10

Some $1$ e $1$ .

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

Etapa 3.4

Simplifique o numerador.

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Etapa 3.4.1

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

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Etapa 3.4.1.1

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

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

Etapa 3.4.1.2

Fatore ${\sin (x)}^{3}$ de $- {\sin (x)}^{2}$ .

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

Etapa 3.4.1.3

Fatore ${\sin (x)}^{3}$ de ${\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) + {\sin (x)}^{3} (- {\sin (x)}^{- 1})$ .

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

Etapa 3.4.2

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

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Etapa 3.4.2.1

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

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

Etapa 3.4.2.2

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

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

Etapa 3.4.2.3

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

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

Etapa 3.4.3

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

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Etapa 3.4.3.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 3.4.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 3.4.3.3

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

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

Etapa 3.4.3.4

Some $1$ e $1$ .

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

Etapa 3.4.4

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

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Etapa 3.4.4.1

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

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

Etapa 3.4.4.2

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

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

Etapa 3.4.4.3

Fatore $\sin (x)$ de $\sin (x) (\cos (x) \sin (x)) + \sin (x) \cdot - 1$ .

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

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

Etapa 3.4.5

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

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Etapa 3.4.5.1

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

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Etapa 3.4.5.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 3.4.5.1.2

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

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

Etapa 3.4.5.2

Some $- 2$ e $1$ .

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

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

Etapa 3.4.6

Multiplique ${\sin (x)}^{3}$ por ${\sin (x)}^{- 1}$ somando os expoentes.

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Etapa 3.4.6.1

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

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

Etapa 3.4.6.2

Subtraia $1$ de $3$ .

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

Etapa 3.5

Simplifique o denominador.

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Etapa 3.5.1

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

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Etapa 3.5.1.1

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

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

Etapa 3.5.1.2

Fatore ${\cos (x)}^{3}$ de $- {\cos (x)}^{2}$ .

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

Etapa 3.5.1.3

Fatore ${\cos (x)}^{3}$ de ${\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) + {\cos (x)}^{3} (- {\cos (x)}^{- 1})$ .

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

Etapa 3.5.2

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

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Etapa 3.5.2.1

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

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

Etapa 3.5.2.2

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

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

Etapa 3.5.2.3

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

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

Etapa 3.5.3

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

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Etapa 3.5.3.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.5.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.5.3.3

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

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

Etapa 3.5.3.4

Some $1$ e $1$ .

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

Etapa 3.5.4

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

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Etapa 3.5.4.1

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

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

Etapa 3.5.4.2

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

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

Etapa 3.5.4.3

Fatore $\cos (x)$ de $\cos (x) (\cos (x) \sin (x)) + \cos (x) \cdot - 1$ .

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

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

Etapa 3.5.5

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

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Etapa 3.5.5.1

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

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Etapa 3.5.5.1.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.5.5.1.2

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

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

Etapa 3.5.5.2

Some $- 2$ e $1$ .

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

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

Etapa 3.5.6

Multiplique ${\cos (x)}^{3}$ por ${\cos (x)}^{- 1}$ somando os expoentes.

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Etapa 3.5.6.1

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

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

Etapa 3.5.6.2

Subtraia $1$ de $3$ .

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

Etapa 3.6

Cancele o fator comum de $\cos (x) \sin (x) - 1$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)}$

Etapa 4

Reescreva $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ como $\tan^{2} (x)$ .

$\tan^{2} (x)$

Etapa 5

Como os dois lados demonstraram ser equivalentes, a equação é uma identidade.

$\frac{\sin^{2} (x) - \tan (x)}{\cos^{2} (x) - \cot (x)} = \tan^{2} (x)$ é uma identidade