Trigonometría Ejemplos

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

Step 1

Simplifica el numerador.

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Aplica la razón del ángulo doble sinusoidal.

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

Usa la razón del ángulo doble para transformar $\cos (2 x)$ a $2 \cos^{2} (x) - 1$ .

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

Resta $1$ de $1$ .

$\frac{0 + 2 \sin (x) \cos (x) + 2 \cos^{2} (x)}{1 + \sin (2 x) - \cos (2 x)}$

Suma $0$ y $2 \sin (x) \cos (x)$ .

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

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

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Factoriza $2 \cos (x)$ de $2 \sin (x) \cos (x)$ .

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

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

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

Factoriza $2 \cos (x)$ de $2 \cos (x) (\sin (x)) + 2 \cos (x) (\cos (x))$ .

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

Step 2

Simplifica el denominador.

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Aplica la razón del ángulo doble sinusoidal.

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

Usa la razón del ángulo doble para transformar $\cos (2 x)$ a $2 \cos^{2} (x) - 1$ .

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

Aplica la propiedad distributiva.

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

Multiplica $2$ por $- 1$ .

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

Multiplica $- 1$ por $- 1$ .

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

Suma $1$ y $1$ .

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

Factoriza $2$ de $2 + 2 \sin (x) \cos (x) - 2 \cos^{2} (x)$ .

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Factoriza $2$ de $2$ .

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

Factoriza $2$ de $2 \sin (x) \cos (x)$ .

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

Factoriza $2$ de $- 2 \cos^{2} (x)$ .

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

Factoriza $2$ de $2 (1) + 2 (\sin (x) \cos (x))$ .

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

Factoriza $2$ de $2 (1 + \sin (x) \cos (x)) + 2 (- \cos^{2} (x))$ .

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

Mueve $- \cos^{2} (x)$ .

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

Aplica la identidad pitagórica.

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

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

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

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

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

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

Factoriza $\sin (x)$ de $\sin (x) \sin (x) + \sin (x) (\cos (x))$ .

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

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

Step 3

Reduce la expresión mediante la cancelación de los factores comunes.

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Cancela el factor común de $2$ .

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Cancela el factor común.

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

Reescribe la expresión.

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

Cancela el factor común de $\sin (x) + \cos (x)$ .

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Cancela el factor común.

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

Reescribe la expresión.

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

Step 4

Convierte de $\frac{\cos (x)}{\sin (x)}$ a $\cot (x)$ .

$\cot (x)$