Trigonometría Ejemplos

$\frac{1 - \sin (t)}{1 + \sin (t)} = {(\sec (t) - \tan (t))}^{2}$

Step 1

Comienza por el lado derecho.

${(\sec (t) - \tan (t))}^{2}$

Step 2

Convierte a senos y cosenos.

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Aplica la identidad recíproca a $\sec (t)$ .

${(\frac{1}{\cos (t)} - \tan (t))}^{2}$

Escribe $\tan (t)$ en senos y cosenos mediante la identidad del cociente.

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

Simplifica.

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Reescribe ${(\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)})}^{2}$ como $(\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)}) (\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)})$ .

$(\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)}) (\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)})$

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

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Aplica la propiedad distributiva.

$\frac{1}{\cos (t)} (\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)}) - \frac{\sin (t)}{\cos (t)} (\frac{1}{\cos (t)} - \frac{\sin (t)}{\cos (t)})$

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Simplifica y combina los términos similares.

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Simplifica cada término.

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Multiplica $- \frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\cos (t)}$ .

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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Multiplica $- 1$ por $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Resta $\frac{\sin (t)}{{\cos (t)}^{2}}$ de $- \frac{\sin (t)}{{\cos (t)}^{2}}$ .

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

Simplifica cada término.

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Combina $- 2$ y $\frac{\sin (t)}{{\cos (t)}^{2}}$ .

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

Mueve el negativo al frente de la fracción.

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

Combina los numeradores sobre el denominador común.

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

Step 3

Simplifica la expresión.

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

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Factoriza $\sin (t)$ de $- 2 \sin (t)$ .

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

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

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

Factoriza $\sin (t)$ de $\sin (t) \cdot - 2 + \sin (t) \sin (t)$ .

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

Combina los numeradores sobre el denominador común.

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

Simplifica el numerador.

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Aplica la propiedad distributiva.

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

Mueve $- 2$ a la izquierda de $\sin (t)$ .

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

Multiplica $\sin (t) \sin (t)$ .

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Factoriza con la regla del cuadrado perfecto.

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

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

Comprueba que el término medio sea dos veces el producto de los números que se elevan al cuadrado en el primer término y el tercer término.

$2 \sin (t) = 2 \cdot 1 \cdot \sin (t)$

Reescribe el polinomio.

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

Factoriza con la regla del trinomio cuadrado perfecto $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , donde $a = 1$ y $b = \sin (t)$ .

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

Step 4

Aplica la identidad pitagórica al revés.

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

Step 5

Simplifica.

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Simplifica el denominador.

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

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

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 $b = \sin (t)$ .

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

Cancela el factor común de ${(1 - \sin (t))}^{2}$ y $1 - \sin (t)$ .

$\frac{1 - \sin (t)}{1 + \sin (t)}$

Step 6

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

$\frac{1 - \sin (t)}{1 + \sin (t)} = {(\sec (t) - \tan (t))}^{2}$ es una identidad