Trigonometría Ejemplos

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)} = \sin (x)$

Step 1

Comienza por el lado izquierdo.

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)}$

Step 2

Simplifica la expresión.

Toca para ver más pasos...

Simplifica el numerador.

Toca para ver más pasos...

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

$\frac{(\frac{1}{\cos (x)} + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)}$

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

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

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

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

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

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

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

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

Multiplica el numerador por la recíproca del denominador.

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

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

Toca para ver más pasos...

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Combina los términos opuestos en $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) + \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})$ .

Toca para ver más pasos...

Reordena los factores en los términos $\frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})$ y $\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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

Suma $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ y $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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

Suma $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ y $0$ .

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

Simplifica cada término.

Toca para ver más pasos...

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

Toca para ver más pasos...

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Reescribe con la propiedad conmutativa de la multiplicación.

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

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

Toca para ver más pasos...

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Combina los numeradores sobre el denominador común.

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

Aplica la identidad pitagórica.

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

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

Toca para ver más pasos...

Cancela el factor común.

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

Reescribe la expresión.

$1 \sin (x)$

Multiplica $\sin (x)$ por $1$ .

$\sin (x)$

Step 3

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

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)} = \sin (x)$ es una identidad