Trigonometria Exemplos

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

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

Etapa 1

Eleve

\cos (x) + \sin (x)

$\cos (x) + \sin (x)$ à potência de

1

$1$ .

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

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

Etapa 2

Eleve

\cos (x) + \sin (x)

$\cos (x) + \sin (x)$ à potência de

1

$1$ .

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

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

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

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

Etapa 4

Some

1

$1$ e

1

$1$ .

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

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

Etapa 5

Reescreva

1

$1$ como

1^{2}

$1^{2}$ .

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

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

Etapa 6

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ em que

a = \cos (x) + \sin (x)

$a = \cos (x) + \sin (x)$ e

b = 1

$b = 1$ .

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

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