Pré-cálculo Exemplos

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

Etapa 1

Comece do lado esquerdo.

$\frac{\cot (x) - \tan (x)}{\cot (x) + \tan (x)}$

Etapa 2

Simplifique a expressão.

Toque para ver mais passagens...

Etapa 2.1

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 2.1.1

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

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

Etapa 2.1.2

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

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

Etapa 2.2

Simplifique o denominador.

Toque para ver mais passagens...

Etapa 2.2.1

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

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

Etapa 2.2.2

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

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

Etapa 2.3

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

Toque para ver mais passagens...

Etapa 2.3.1

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

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

Etapa 2.3.2

Combine.

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

Etapa 2.4

Aplique a propriedade distributiva.

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

Etapa 2.5

Simplifique cancelando.

Toque para ver mais passagens...

Etapa 2.5.1

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

Toque para ver mais passagens...

Etapa 2.5.1.1

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

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

Etapa 2.5.1.2

Cancele o fator comum.

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

Etapa 2.5.1.3

Reescreva a expressão.

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

Etapa 2.5.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 2.5.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 2.5.4

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

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

Etapa 2.5.5

Some $1$ e $1$ .

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

Etapa 2.5.6

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

Toque para ver mais passagens...

Etapa 2.5.6.1

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

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

Etapa 2.5.6.2

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

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

Etapa 2.5.6.3

Cancele o fator comum.

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

Etapa 2.5.6.4

Reescreva a expressão.

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

Etapa 2.5.7

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.8

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.9

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

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

Etapa 2.5.10

Some $1$ e $1$ .

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

Etapa 2.5.11

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

Toque para ver mais passagens...

Etapa 2.5.11.1

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

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

Etapa 2.5.11.2

Cancele o fator comum.

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

Etapa 2.5.11.3

Reescreva a expressão.

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

Etapa 2.5.12

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 2.5.13

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 2.5.14

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

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

Etapa 2.5.15

Some $1$ e $1$ .

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

Etapa 2.5.16

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

Toque para ver mais passagens...

Etapa 2.5.16.1

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

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

Etapa 2.5.16.2

Cancele o fator comum.

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

Etapa 2.5.16.3

Reescreva a expressão.

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

Etapa 2.5.17

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.18

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 2.5.19

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

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

Etapa 2.5.20

Some $1$ e $1$ .

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

Etapa 2.6

Reorganize os termos.

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

Etapa 2.7

Aplique a identidade trigonométrica fundamental.

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

Etapa 2.8

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

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

Etapa 2.9

Aplique a fórmula do arco duplo do cosseno.

$\cos (2 x)$

Etapa 3

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

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