Trigonometria Exemplos

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$

Etapa 1

Comece do lado esquerdo.

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)}$

Etapa 2

Converta em senos e cossenos.

Toque para ver mais passagens...

Etapa 2.1

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

$\frac{\frac{\sin (a)}{\cos (a)} - \tan (b)}{\cot (b) - \cot (a)}$

Etapa 2.2

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

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\cot (b) - \cot (a)}$

Etapa 2.3

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

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \cot (a)}$

Etapa 2.4

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

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$

Etapa 3

Simplifique.

Toque para ver mais passagens...

Etapa 3.1

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

Toque para ver mais passagens...

Etapa 3.1.1

Multiplique $\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$ por $\frac{\cos (a) \cos (b) \sin (b) \sin (a)}{\cos (a) \cos (b) \sin (b) \sin (a)}$ .

$\frac{\cos (a) \cos (b) \sin (b) \sin (a)}{\cos (a) \cos (b) \sin (b) \sin (a)} \cdot \frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$

Etapa 3.1.2

Combine.

$\frac{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)})}$

Etapa 3.2

Aplique a propriedade distributiva.

$\frac{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\sin (a)}{\cos (a)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3

Simplifique cancelando.

Toque para ver mais passagens...

Etapa 3.3.1

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

Toque para ver mais passagens...

Etapa 3.3.1.1

Fatore $\cos (a)$ de $\cos (a) \cos (b) \sin (b) \sin (a)$ .

$\frac{\cos (a) (\cos (b) \sin (b) \sin (a)) \frac{\sin (a)}{\cos (a)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3.1.2

Cancele o fator comum.

Etapa 3.3.1.3

Reescreva a expressão.

$\frac{\cos (b) \sin (b) \sin (a) \sin (a) + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3.2

Eleve $\sin (a)$ à potência de $1$ .

$\frac{\cos (b) \sin (b) ({\sin (a)}^{1} \sin (a)) + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3.3

Eleve $\sin (a)$ à potência de $1$ .

$\frac{\cos (b) \sin (b) ({\sin (a)}^{1} {\sin (a)}^{1}) + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3.4

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

$\frac{\cos (b) \sin (b) {\sin (a)}^{1 + 1} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Etapa 3.3.5

Some $1$ e $1$ .

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

Etapa 3.3.6

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

Toque para ver mais passagens...

Etapa 3.3.6.1

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

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

Etapa 3.3.6.2

Fatore $\cos (b)$ de $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Etapa 3.3.6.3

Cancele o fator comum.

Etapa 3.3.6.4

Reescreva a expressão.

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

Etapa 3.3.7

Eleve $\sin (b)$ à potência de $1$ .

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

Etapa 3.3.8

Eleve $\sin (b)$ à potência de $1$ .

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

Etapa 3.3.9

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

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

Etapa 3.3.10

Some $1$ e $1$ .

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

Etapa 3.3.11

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

Toque para ver mais passagens...

Etapa 3.3.11.1

Fatore $\sin (b)$ de $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Etapa 3.3.11.2

Cancele o fator comum.

Etapa 3.3.11.3

Reescreva a expressão.

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

Etapa 3.3.12

Eleve $\cos (b)$ à potência de $1$ .

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

Etapa 3.3.13

Eleve $\cos (b)$ à potência de $1$ .

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

Etapa 3.3.14

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

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

Etapa 3.3.15

Some $1$ e $1$ .

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

Etapa 3.3.16

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

Toque para ver mais passagens...

Etapa 3.3.16.1

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

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

Etapa 3.3.16.2

Fatore $\sin (a)$ de $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Etapa 3.3.16.3

Cancele o fator comum.

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

Etapa 3.3.16.4

Reescreva a expressão.

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

Etapa 3.3.17

Eleve $\cos (a)$ à potência de $1$ .

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

Etapa 3.3.18

Eleve $\cos (a)$ à potência de $1$ .

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

Etapa 3.3.19

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

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

Etapa 3.3.20

Some $1$ e $1$ .

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

Etapa 3.4

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

Toque para ver mais passagens...

Etapa 3.4.1

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

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

Etapa 3.4.2

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

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

Etapa 3.4.3

Fatore $\sin (b) \sin (a)$ de $\sin (b) \sin (a) (\cos (b) \sin (a)) + \sin (b) \sin (a) (\cos (a) (- \sin (b)))$ .

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

Etapa 3.5

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

Toque para ver mais passagens...

Etapa 3.5.1

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

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

Etapa 3.5.2

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

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) + \cos (a) (- \sin (b)))}{\cos (a) \cos (b) (\cos (b) \sin (a)) + \cos (a) \cos (b) (\sin (b) (- \cos (a)))}$

Etapa 3.5.3

Fatore $\cos (a) \cos (b)$ de $\cos (a) \cos (b) (\cos (b) \sin (a)) + \cos (a) \cos (b) (\sin (b) (- \cos (a)))$ .

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) + \cos (a) (- \sin (b)))}{\cos (a) \cos (b) (\cos (b) \sin (a) + \sin (b) (- \cos (a)))}$

Etapa 3.6

Reordene os termos.

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) - 1 \cdot \cos (a) \sin (b))}{\cos (a) \cos (b) (\cos (b) \sin (a) - 1 \cdot \sin (b) \cos (a))}$

Etapa 3.7

Cancele o fator comum de $\cos (b) \sin (a) - 1 \cdot \cos (a) \sin (b)$ e $\cos (b) \sin (a) - 1 \cdot \sin (b) \cos (a)$ .

$\frac{\sin (b) \sin (a)}{\cos (a) \cos (b)}$

Etapa 4

Reordene os termos.

$\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$

Etapa 5

Reescreva $\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$ como $\frac{\tan (b)}{\cot (a)}$ .

$\frac{\tan (b)}{\cot (a)}$

Etapa 6

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

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$ é uma identidade