Trigonometria Exemplos

$\csc (x) - \cot (x) = \frac{1}{\csc (x) + \cot (x)}$

Etapa 1

Comece do lado direito.

$\frac{1}{\csc (x) + \cot (x)}$

Etapa 2

Converta em senos e cossenos.

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Etapa 2.1

Aplique a identidade recíproca a $\csc (x)$ .

$\frac{1}{\frac{1}{\sin (x)} + \cot (x)}$

Etapa 2.2

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

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

Etapa 3

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

$\frac{1}{\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}} \cdot \frac{\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}$

Etapa 4

Combine.

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

Etapa 5

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

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

Etapa 6

Simplifique o denominador.

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Etapa 6.1

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

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Etapa 6.1.1

Aplique a propriedade distributiva.

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

Etapa 6.1.2

Aplique a propriedade distributiva.

$\frac{\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} (\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)})}$

Etapa 6.1.3

Aplique a propriedade distributiva.

Etapa 6.2

Simplifique e combine termos semelhantes.

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Etapa 6.2.1

Simplifique cada termo.

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Etapa 6.2.1.1

Multiplique $\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Etapa 6.2.1.1.1

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

$\frac{\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x) \sin (x)} + \frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}}$

Etapa 6.2.1.1.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.1.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.1.4

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

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

Etapa 6.2.1.1.5

Some $1$ e $1$ .

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

Etapa 6.2.1.2

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

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Etapa 6.2.1.2.1

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

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

Etapa 6.2.1.2.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.2.4

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

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

Etapa 6.2.1.2.5

Some $1$ e $1$ .

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

Etapa 6.2.1.3

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

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Etapa 6.2.1.3.1

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

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

Etapa 6.2.1.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.3.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.3.4

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

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

Etapa 6.2.1.3.5

Some $1$ e $1$ .

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

Etapa 6.2.1.4

Multiplique $\frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ .

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Etapa 6.2.1.4.1

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

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

Etapa 6.2.1.4.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.4.3

Eleve $\cos (x)$ à potência de $1$ .

Etapa 6.2.1.4.4

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

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

Etapa 6.2.1.4.5

Some $1$ e $1$ .

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

Etapa 6.2.1.4.6

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.4.7

Eleve $\sin (x)$ à potência de $1$ .

Etapa 6.2.1.4.8

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

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

Etapa 6.2.1.4.9

Some $1$ e $1$ .

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

Etapa 6.2.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 6.3

Combine os numeradores em relação ao denominador comum.

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

Etapa 6.4

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

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

Etapa 6.5

Fatore usando a regra do quadrado perfeito.

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

Etapa 7

Aplique a identidade trigonométrica fundamental inversa.

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

Etapa 8

Simplifique.

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Etapa 8.1

Multiplique o numerador pelo inverso do denominador.

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

Etapa 8.2

Simplifique o numerador.

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Etapa 8.2.1

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

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

Etapa 8.2.2

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 $b = \cos (x)$ .

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

Etapa 8.3

Cancele os fatores comuns.

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Etapa 8.3.1

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

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

Etapa 8.3.2

Cancele o fator comum.

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

Etapa 8.3.3

Reescreva a expressão.

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

Etapa 8.4

Aplique a propriedade distributiva.

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

Etapa 8.5

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

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

Etapa 8.6

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

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

Etapa 8.7

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.8

Simplifique cada termo.

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Etapa 8.8.1

Aplique a propriedade distributiva.

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

Etapa 8.8.2

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

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

Etapa 8.8.3

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

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Etapa 8.8.3.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 8.8.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 8.8.3.3

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

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

Etapa 8.8.3.4

Some $1$ e $1$ .

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

Etapa 8.9

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

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

Etapa 8.10

Some $1$ e $0$ .

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

Etapa 8.11

Simplifique o numerador.

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Etapa 8.11.1

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

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

Etapa 8.11.2

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 $b = \cos (x)$ .

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

Etapa 8.12

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

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

Etapa 9

Agora, considere o lado esquerdo da equação.

$\csc (x) - \cot (x)$

Etapa 10

Converta em senos e cossenos.

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Etapa 10.1

Aplique a identidade recíproca a $\csc (x)$ .

$\frac{1}{\sin (x)} - \cot (x)$

Etapa 10.2

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

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

Etapa 11

Combine os numeradores em relação ao denominador comum.

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

Etapa 12

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

$\csc (x) - \cot (x) = \frac{1}{\csc (x) + \cot (x)}$ é uma identidade