Álgebra Exemplos

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$

Etapa 1

Para escrever $\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)}$ como fração com um denominador comum, multiplique por $\frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)}$ .

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} \cdot \frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$

Etapa 2

Para escrever $\frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$ como fração com um denominador comum, multiplique por $\frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$ .

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} \cdot \frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)} \cdot \frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$

Etapa 3

Escreva cada expressão com um denominador comum de $(\sin (A) - \sin (B)) (\cos (A) - \cos (B))$ , multiplicando cada um por um fator apropriado de $1$ .

Toque para ver mais passagens...

Etapa 3.1

Multiplique $\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)}$ por $\frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)}$ .

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\sin (A) - \sin (B)) (\cos (A) - \cos (B))} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)} \cdot \frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$

Etapa 3.2

Multiplique $\frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$ por $\frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$ .

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\sin (A) - \sin (B)) (\cos (A) - \cos (B))} + \frac{(\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 3.3

Reordene os fatores de $(\sin (A) - \sin (B)) (\cos (A) - \cos (B))$ .

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))} + \frac{(\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 4

Combine os numeradores em relação ao denominador comum.

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 5.1

Expanda $(\cos (A) + \cos (B)) (\cos (A) - \cos (B))$ usando o método FOIL.

Toque para ver mais passagens...

Etapa 5.1.1

Aplique a propriedade distributiva.

$\frac{\cos (A) (\cos (A) - \cos (B)) + \cos (B) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.1.2

Aplique a propriedade distributiva.

$\frac{\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.1.3

Aplique a propriedade distributiva.

$\frac{\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.2

Combine os termos opostos em $\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) \cos (A) + \cos (B) (- \cos (B))$ .

Toque para ver mais passagens...

Etapa 5.2.1

Reorganize os fatores nos termos $\cos (A) (- \cos (B))$ e $\cos (B) \cos (A)$ .

$\frac{\cos (A) \cos (A) - \cos (A) \cos (B) + \cos (A) \cos (B) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.2.2

Some $- \cos (A) \cos (B)$ e $\cos (A) \cos (B)$ .

$\frac{\cos (A) \cos (A) + 0 + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.2.3

Some $\cos (A) \cos (A)$ e $0$ .

$\frac{\cos (A) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.3

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 5.3.1

Multiplique $\cos (A) \cos (A)$ .

Toque para ver mais passagens...

Etapa 5.3.1.1

Eleve $\cos (A)$ à potência de $1$ .

$\frac{\cos^{1} (A) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.3.1.2

Eleve $\cos (A)$ à potência de $1$ .

$\frac{\cos^{1} (A) \cos^{1} (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.3.1.3

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

$\frac{{\cos (A)}^{1 + 1} + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.3.1.4

Some $1$ e $1$ .

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

Etapa 5.3.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 5.3.3

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

Toque para ver mais passagens...

Etapa 5.3.3.1

Eleve $\cos (B)$ à potência de $1$ .

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

Etapa 5.3.3.2

Eleve $\cos (B)$ à potência de $1$ .

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

Etapa 5.3.3.3

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

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

Etapa 5.3.3.4

Some $1$ e $1$ .

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

Etapa 5.4

Expanda $(\sin (A) + \sin (B)) (\sin (A) - \sin (B))$ usando o método FOIL.

Toque para ver mais passagens...

Etapa 5.4.1

Aplique a propriedade distributiva.

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

Etapa 5.4.2

Aplique a propriedade distributiva.

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

Etapa 5.4.3

Aplique a propriedade distributiva.

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

Etapa 5.5

Combine os termos opostos em $\sin (A) \sin (A) + \sin (A) (- \sin (B)) + \sin (B) \sin (A) + \sin (B) (- \sin (B))$ .

Toque para ver mais passagens...

Etapa 5.5.1

Reorganize os fatores nos termos $\sin (A) (- \sin (B))$ e $\sin (B) \sin (A)$ .

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

Etapa 5.5.2

Some $- \sin (A) \sin (B)$ e $\sin (A) \sin (B)$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) + 0 + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.5.3

Some $\sin (A) \sin (A)$ e $0$ .

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

Etapa 5.6

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 5.6.1

Multiplique $\sin (A) \sin (A)$ .

Toque para ver mais passagens...

Etapa 5.6.1.1

Eleve $\sin (A)$ à potência de $1$ .

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

Etapa 5.6.1.2

Eleve $\sin (A)$ à potência de $1$ .

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

Etapa 5.6.1.3

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

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

Etapa 5.6.1.4

Some $1$ e $1$ .

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

Etapa 5.6.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 5.6.3

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

Toque para ver mais passagens...

Etapa 5.6.3.1

Eleve $\sin (B)$ à potência de $1$ .

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

Etapa 5.6.3.2

Eleve $\sin (B)$ à potência de $1$ .

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

Etapa 5.6.3.3

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

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

Etapa 5.6.3.4

Some $1$ e $1$ .

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

Etapa 5.7

Reescreva $\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - \sin^{2} (B)$ em uma forma fatorada.

Toque para ver mais passagens...

Etapa 5.7.1

Reagrupe os termos.

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

Etapa 5.7.2

Reorganize os termos.

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

Etapa 5.7.3

Aplique a identidade trigonométrica fundamental.

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

Etapa 5.7.4

Reescreva $1 - \sin^{2} (B)$ em uma forma fatorada.

Toque para ver mais passagens...

Etapa 5.7.4.1

Reescreva $1$ como $1^{2}$ .

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

Etapa 5.7.4.2

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados, $a^{2} - b^{2} = (a + b) (a - b)$ em que $a = 1$ e $b = \sin (B)$ .

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

Etapa 5.7.5

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

Toque para ver mais passagens...

Etapa 5.7.5.1

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

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

Etapa 5.7.5.2

Fatore $- 1$ de $(1 + \sin (B)) (1 - \sin (B))$ .

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

Etapa 5.7.5.3

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

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

Etapa 5.7.6

Expanda $(- 1 - \sin (B)) (1 - \sin (B))$ usando o método FOIL.

Toque para ver mais passagens...

Etapa 5.7.6.1

Aplique a propriedade distributiva.

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

Etapa 5.7.6.2

Aplique a propriedade distributiva.

$\frac{- (\cos^{2} (B) - 1 \cdot 1 - 1 (- \sin (B)) - \sin (B) (1 - \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.6.3

Aplique a propriedade distributiva.

$\frac{- (\cos^{2} (B) - 1 \cdot 1 - 1 (- \sin (B)) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.7

Simplifique e combine termos semelhantes.

Toque para ver mais passagens...

Etapa 5.7.7.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 5.7.7.1.1

Multiplique $- 1$ por $1$ .

$\frac{- (\cos^{2} (B) - 1 - 1 (- \sin (B)) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.7.1.2

Multiplique $- 1 (- \sin (B))$ .

Toque para ver mais passagens...

Etapa 5.7.7.1.2.1

Multiplique $- 1$ por $- 1$ .

$\frac{- (\cos^{2} (B) - 1 + 1 \sin (B) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.7.1.2.2

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.7.1.3

Multiplique $- 1$ por $1$ .

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

Etapa 5.7.7.1.4

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

Toque para ver mais passagens...

Etapa 5.7.7.1.4.1

Multiplique $- 1$ por $- 1$ .

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

Etapa 5.7.7.1.4.2

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

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

Etapa 5.7.7.1.4.3

Eleve $\sin (B)$ à potência de $1$ .

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

Etapa 5.7.7.1.4.4

Eleve $\sin (B)$ à potência de $1$ .

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

Etapa 5.7.7.1.4.5

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

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

Etapa 5.7.7.1.4.6

Some $1$ e $1$ .

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

Etapa 5.7.7.2

Subtraia $\sin (B)$ de $\sin (B)$ .

$\frac{- (\cos^{2} (B) - 1 + 0 + \sin^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 5.7.7.3

Some $- 1$ e $0$ .

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

Etapa 5.7.8

Reescreva $- 1$ como $- 1 (1)$ .

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

Etapa 5.7.9

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

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

Etapa 5.7.10

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

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

Etapa 5.7.11

Reescreva $- 1 (1 - \sin^{2} (B))$ como $- (1 - \sin^{2} (B))$ .

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

Etapa 5.7.12

Aplique a identidade trigonométrica fundamental.

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

Etapa 5.7.13

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

$\frac{- 0}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 6

Multiplique por zero.

Toque para ver mais passagens...

Etapa 6.1

Multiplique $- 1$ por $0$ .

$\frac{0}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Etapa 6.2

Divida $0$ por $(\cos (A) - \cos (B)) (\sin (A) - \sin (B))$ .

$0$