Pré-cálculo Exemplos

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

Etapa 1

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 1.1

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

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

Etapa 1.2

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

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

Etapa 1.3

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

Toque para ver mais passagens...

Etapa 1.3.1

Aplique a propriedade distributiva.

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

Etapa 1.3.2

Aplique a propriedade distributiva.

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

Etapa 1.3.3

Aplique a propriedade distributiva.

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

Etapa 1.4

Simplifique e combine termos semelhantes.

Toque para ver mais passagens...

Etapa 1.4.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.4.1.1

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

Toque para ver mais passagens...

Etapa 1.4.1.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.1.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.1.3

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

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

Etapa 1.4.1.1.4

Some $1$ e $1$ .

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

Etapa 1.4.1.2

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

Toque para ver mais passagens...

Etapa 1.4.1.2.1

Combine $\sin (x)$ e $\frac{\sin (x)}{\cos (x)}$ .

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

Etapa 1.4.1.2.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.2.4

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

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

Etapa 1.4.1.2.5

Some $1$ e $1$ .

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

Etapa 1.4.1.3

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

Toque para ver mais passagens...

Etapa 1.4.1.3.1

Combine $\frac{\sin (x)}{\cos (x)}$ e $\sin (x)$ .

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

Etapa 1.4.1.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.3.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.3.4

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

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

Etapa 1.4.1.3.5

Some $1$ e $1$ .

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

Etapa 1.4.1.4

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

Toque para ver mais passagens...

Etapa 1.4.1.4.1

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

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

Etapa 1.4.1.4.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.4.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.4.1.4.4

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

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

Etapa 1.4.1.4.5

Some $1$ e $1$ .

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

Etapa 1.4.1.4.6

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.4.1.4.7

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.4.1.4.8

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

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

Etapa 1.4.1.4.9

Some $1$ e $1$ .

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

Etapa 1.4.2

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

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

Etapa 1.5

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

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

Etapa 1.6

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

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

Etapa 1.7

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

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

Etapa 1.8

Um elevado a qualquer potência é um.

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

Etapa 1.9

Reescreva $\sin^{2} (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x) - \frac{1}{\cos^{2} (x)}$ em uma forma fatorada.

Toque para ver mais passagens...

Etapa 1.9.1

Deixe $u = \frac{2 \sin^{2} (x)}{\cos (x)}$ . Substitua $u$ em todas as ocorrências de $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

Toque para ver mais passagens...

Etapa 1.9.1.1

Subtraia $1$ de $1$ .

$\frac{u + 0}{\tan (x)}$

Etapa 1.9.1.2

Some $u$ e $0$ .

$\frac{u}{\tan (x)}$

Etapa 1.9.2

Substitua todas as ocorrências de $u$ por $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

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

Etapa 2

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

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

Etapa 3

Multiplique o numerador pelo inverso do denominador.

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

Etapa 4

Combine.

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

Etapa 5

Cancele o fator comum de $\sin^{2} (x)$ e $\sin (x)$ .

Toque para ver mais passagens...

Etapa 5.1

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

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

Etapa 5.2

Cancele os fatores comuns.

Toque para ver mais passagens...

Etapa 5.2.1

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

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

Etapa 5.2.2

Cancele o fator comum.

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

Etapa 5.2.3

Reescreva a expressão.

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

Etapa 6

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

Toque para ver mais passagens...

Etapa 6.1

Cancele o fator comum.

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

Etapa 6.2

Divida $2 \sin (x)$ por $1$ .

$2 \sin (x)$