Trigonometría Ejemplos

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

Step 1

Comienza por el lado izquierdo.

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

Step 2

Simplifica la expresión.

Toca para ver más pasos...

Simplifica cada término.

Toca para ver más pasos...

Aplica la identidad pitagórica.

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

Simplifica el numerador.

Toca para ver más pasos...

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

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

Dado que ambos términos son cuadrados perfectos, factoriza con la fórmula de la diferencia de cuadrados, $a^{2} - b^{2} = (a + b) (a - b)$ , donde $a = 1$ y $b = \cot (x)$ .

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

Simplifica.

Toca para ver más pasos...

Reescribe $\cot (x)$ en términos de senos y cosenos.

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

Reescribe $\cot (x)$ en términos de senos y cosenos.

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

Simplifica el denominador.

Toca para ver más pasos...

Reescribe $\csc (x)$ en términos de senos y cosenos.

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

Aplica la regla del producto a $\frac{1}{\sin (x)}$ .

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

Uno elevado a cualquier potencia es uno.

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

Multiplica el numerador por la recíproca del denominador.

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

Expande $(1 + \frac{\cos (x)}{\sin (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ con el método PEIU (primero, exterior, interior, ultimo).

Toca para ver más pasos...

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Simplifica y combina los términos similares.

Toca para ver más pasos...

Simplifica cada término.

Toca para ver más pasos...

Multiplica $1$ por $1$ .

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

Multiplica $- \frac{\cos (x)}{\sin (x)}$ por $1$ .

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

Multiplica $\frac{\cos (x)}{\sin (x)}$ por $1$ .

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

Reescribe con la propiedad conmutativa de la multiplicación.

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

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

Toca para ver más pasos...

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

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

Eleva $\cos (x)$ a la potencia de $1$ .

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

Eleva $\cos (x)$ a la potencia de $1$ .

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

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Suma $1$ y $1$ .

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

Eleva $\sin (x)$ a la potencia de $1$ .

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

Eleva $\sin (x)$ a la potencia de $1$ .

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

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Suma $1$ y $1$ .

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

Suma $- \frac{\cos (x)}{\sin (x)}$ y $\frac{\cos (x)}{\sin (x)}$ .

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

Suma $1$ y $0$ .

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

Aplica la propiedad distributiva.

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

Multiplica $\sin^{2} (x)$ por $1$ .

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

Cancela el factor común de $\sin^{2} (x)$ .

Toca para ver más pasos...

Mueve el signo menos inicial en $- \frac{\cos^{2} (x)}{\sin^{2} (x)}$ al numerador.

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

Cancela el factor común.

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

Reescribe la expresión.

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

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

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

Aplica la identidad pitagórica.

$1$

Step 3

Debido a que se ha demostrado que los dos lados son equivalentes, la ecuación es una identidad.

$\frac{1 - \cot^{2} (x)}{1 + \cot^{2} (x)} + 2 \cos^{2} (x) = 1$ es una identidad