Trigonometria Exemplos

\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$

Etapa 1

Comece do lado direito.

\sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}

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

Etapa 2

Simplifique a expressão.

Toque para ver mais passagens...

Etapa 2.1

Para escrever

\sin (t)

$\sin (t)$ como fração com um denominador comum, multiplique por

\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

\frac{\sin (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)} - \frac{1}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.2

Combine os numeradores em relação ao denominador comum.

\frac{\sin (t) (\sin (t) + \cos (t)) - 1}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.3

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 2.3.1

Aplique a propriedade distributiva.

\frac{\sin (t) \sin (t) + \sin (t) \cos (t) - 1}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.3.2

Multiplique

\sin (t) \sin (t)

$\sin (t) \sin (t)$ .

Toque para ver mais passagens...

Etapa 2.3.2.1

Eleve

\sin (t)

$\sin (t)$ à potência de

1

$1$ .

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

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

Etapa 2.3.2.2

Eleve

\sin (t)

$\sin (t)$ à potência de

1

$1$ .

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

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

Etapa 2.3.2.3

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

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

Etapa 2.3.2.4

Some

1

$1$ e

1

$1$ .

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

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

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

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

Etapa 2.3.3

Mova

- 1

$- 1$ .

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

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

Etapa 2.3.4

Reordene

\sin^{2} (t)

$\sin^{2} (t)$ e

- 1

$- 1$ .

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

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

Etapa 2.3.5

Reescreva

- 1

$- 1$ como

- 1 (1)

$- 1 (1)$ .

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

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

Etapa 2.3.6

Fatore

- 1

$- 1$ de

\sin^{2} (t)

$\sin^{2} (t)$ .

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

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

Etapa 2.3.7

Fatore

- 1

$- 1$ de

- 1 (1) - 1 (- \sin^{2} (t))

$- 1 (1) - 1 (- \sin^{2} (t))$ .

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

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

Etapa 2.3.8

Reescreva

- 1 (1 - \sin^{2} (t))

$- 1 (1 - \sin^{2} (t))$ como

- (1 - \sin^{2} (t))

$- (1 - \sin^{2} (t))$ .

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

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

Etapa 2.3.9

Aplique a identidade trigonométrica fundamental.

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

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

Etapa 2.3.10

Fatore

\cos (t)

$\cos (t)$ de

- \cos^{2} (t) + \sin (t) \cos (t)

$- \cos^{2} (t) + \sin (t) \cos (t)$ .

Toque para ver mais passagens...

Etapa 2.3.10.1

Fatore

\cos (t)

$\cos (t)$ de

- \cos^{2} (t)

$- \cos^{2} (t)$ .

\frac{\cos (t) (- \cos (t)) + \sin (t) \cos (t)}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.3.10.2

Fatore

\cos (t)

$\cos (t)$ de

\sin (t) \cos (t)

$\sin (t) \cos (t)$ .

\frac{\cos (t) (- \cos (t)) + \cos (t) \sin (t)}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.3.10.3

Fatore

\cos (t)

$\cos (t)$ de

\cos (t) (- \cos (t)) + \cos (t) \sin (t)

$\cos (t) (- \cos (t)) + \cos (t) \sin (t)$ .

\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)

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

\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)

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

\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)

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

Etapa 2.4

Para escrever

\cos (t)

$\cos (t)$ como fração com um denominador comum, multiplique por

\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \frac{\cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}

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

Etapa 2.5

Combine os numeradores em relação ao denominador comum.

\frac{\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}

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

Etapa 2.6

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 2.6.1

Fatore

\cos (t)

$\cos (t)$ de

\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))

$\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))$ .

\frac{\cos (t) (- \cos (t) + \sin (t) + \sin (t) + \cos (t))}{\sin (t) + \cos (t)}

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

Etapa 2.6.2

Some

- \cos (t)

$- \cos (t)$ e

\cos (t)

$\cos (t)$ .

\frac{\cos (t) (0 + \sin (t) + \sin (t))}{\sin (t) + \cos (t)}

$\frac{\cos (t) (0 + \sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

Etapa 2.6.3

Some

0

$0$ e

\sin (t)

$\sin (t)$ .

\frac{\cos (t) (\sin (t) + \sin (t))}{\sin (t) + \cos (t)}

$\frac{\cos (t) (\sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

Etapa 2.6.4

Some

\sin (t)

$\sin (t)$ e

\sin (t)

$\sin (t)$ .

\frac{\cos (t) \cdot 2 \sin (t)}{\sin (t) + \cos (t)}

$\frac{\cos (t) \cdot 2 \sin (t)}{\sin (t) + \cos (t)}$

\frac{\cos (t) \cdot 2 \sin (t)}{\sin (t) + \cos (t)}

$\frac{\cos (t) \cdot 2 \sin (t)}{\sin (t) + \cos (t)}$

Etapa 2.7

Mova

2

$2$ para a esquerda de

\cos (t)

$\cos (t)$ .

\frac{2 \cos (t) \sin (t)}{\sin (t) + \cos (t)}

$\frac{2 \cos (t) \sin (t)}{\sin (t) + \cos (t)}$

\frac{2 \cos (t) \sin (t)}{\sin (t) + \cos (t)}

$\frac{2 \cos (t) \sin (t)}{\sin (t) + \cos (t)}$

Etapa 3

Reordene os termos.

\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}

$\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}$

Etapa 4

Reescreva

\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}

$\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}$ como

\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}$ .

\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}$

Etapa 5

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

\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$ é uma identidade