Trigonometría Ejemplos

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

Step 1

Simplifica el numerador.

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

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

Aplica la regla del producto a $\frac{\sin (t)}{\cos (t)}$ .

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

Mueve $- 1$ .

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

Reordena $\cos^{2} (t)$ y $- 1$ .

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

Reescribe $- 1$ como $- 1 (1)$ .

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

Factoriza $- 1$ de $\cos^{2} (t)$ .

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

Factoriza $- 1$ de $- 1 (1) - 1 (- \cos^{2} (t))$ .

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

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

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

Aplica la identidad pitagórica.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reordena $- 1^{2}$ y ${(\frac{1}{\cos (t)})}^{2}$ .

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

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 $a = \frac{1}{\cos (t)}$ y $b = 1$ .

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

Step 2

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

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Cancela el factor común.

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

Divide $(\frac{1}{\cos (t)} + 1) (\frac{1}{\cos (t)} - 1)$ por $1$ .

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

Step 3

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

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Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Step 4

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Combina $\frac{1}{\cos (t)}$ y $- 1$ .

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

Mueve el negativo al frente de la fracción.

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

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

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

Multiplica $- 1$ por $1$ .

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

Suma $- \frac{1}{\cos (t)}$ y $\frac{1}{\cos (t)}$ .

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

Suma $\frac{1}{\cos^{2} (t)}$ y $0$ .

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

Step 5

Simplifica cada término.

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

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

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

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

Convierte de $\frac{1}{\cos (t)}$ a $\sec (t)$ .

$\sec^{2} (t) - 1$

Step 6

Aplica la identidad pitagórica.

$\tan^{2} (t)$