Álgebra Exemplos

$\tan (6 x)$

Etapa 1

Fatore $2$ de $6 x$ .

$\tan (2 (3 x))$

Etapa 2

Simplifique o numerador.

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Etapa 2.1

Aplique a fórmula do arco triplo da tangente.

$\frac{2 \frac{3 \tan (x) - \tan^{3} (x)}{1 - 3 \tan^{2} (x)}}{1 - \tan^{2} (3 x)}$

Etapa 2.2

Fatore $\tan (x)$ de $3 \tan (x) - \tan^{3} (x)$ .

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Etapa 2.2.1

Fatore $\tan (x)$ de $3 \tan (x)$ .

$\frac{2 \frac{\tan (x) \cdot 3 - \tan^{3} (x)}{1 - 3 \tan^{2} (x)}}{1 - \tan^{2} (3 x)}$

Etapa 2.2.2

Fatore $\tan (x)$ de $- \tan^{3} (x)$ .

$\frac{2 \frac{\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{1 - \tan^{2} (3 x)}$

Etapa 2.2.3

Fatore $\tan (x)$ de $\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{1 - \tan^{2} (3 x)}$

Etapa 3

Simplifique o denominador.

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Etapa 3.1

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

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{1^{2} - \tan^{2} (3 x)}$

Etapa 3.2

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 $b = \tan (3 x)$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \tan (3 x)) (1 - \tan (3 x))}$

Etapa 3.3

Simplifique.

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Etapa 3.3.1

Aplique a fórmula do arco triplo da tangente.

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{3 \tan (x) - \tan^{3} (x)}{1 - 3 \tan^{2} (x)}) (1 - \tan (3 x))}$

Etapa 3.3.2

Fatore $\tan (x)$ de $3 \tan (x) - \tan^{3} (x)$ .

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Etapa 3.3.2.1

Fatore $\tan (x)$ de $3 \tan (x)$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) \cdot 3 - \tan^{3} (x)}{1 - 3 \tan^{2} (x)}) (1 - \tan (3 x))}$

Etapa 3.3.2.2

Fatore $\tan (x)$ de $- \tan^{3} (x)$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \tan (3 x))}$

Etapa 3.3.2.3

Fatore $\tan (x)$ de $\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \tan (3 x))}$

Etapa 3.3.3

Aplique a fórmula do arco triplo da tangente.

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{3 \tan (x) - \tan^{3} (x)}{1 - 3 \tan^{2} (x)})}$

Etapa 3.3.4

Fatore $\tan (x)$ de $3 \tan (x) - \tan^{3} (x)$ .

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Etapa 3.3.4.1

Fatore $\tan (x)$ de $3 \tan (x)$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) \cdot 3 - \tan^{3} (x)}{1 - 3 \tan^{2} (x)})}$

Etapa 3.3.4.2

Fatore $\tan (x)$ de $- \tan^{3} (x)$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$

Etapa 3.3.4.3

Fatore $\tan (x)$ de $\tan (x) \cdot 3 + \tan (x) (- \tan^{2} (x))$ .

$\frac{2 \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$

Etapa 4

Combine $2$ e $\frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}$ .

$\frac{\frac{2 (\tan (x) (3 - \tan^{2} (x)))}{1 - 3 \tan^{2} (x)}}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$

Etapa 5

Multiplique o numerador pelo inverso do denominador.

$\frac{2 \tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)} \cdot \frac{1}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$

Etapa 6

Multiplique $\frac{2 \tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}$ por $\frac{1}{(1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$ .

$\frac{2 \tan (x) (3 - \tan^{2} (x))}{(1 - 3 \tan^{2} (x)) (1 + \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)}) (1 - \frac{\tan (x) (3 - \tan^{2} (x))}{1 - 3 \tan^{2} (x)})}$