Trigonometria Exemplos

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} = \cos^{4} (w)$

Etapa 1

Subtraia $\cos^{4} (w)$ dos dois lados da equação.

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2

Simplifique $\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w)$ .

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Etapa 2.1

Simplifique cada termo.

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Etapa 2.1.1

Simplifique o numerador.

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Etapa 2.1.1.1

Reescreva $\cot (w)$ em termos de senos e cossenos.

$\frac{- 1 + {(\frac{\cos (w)}{\sin (w)})}^{2} + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.2

Aplique a regra do produto a $\frac{\cos (w)}{\sin (w)}$ .

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.3

Reescreva $\tan (w)$ em termos de senos e cossenos.

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) {(\frac{\sin (w)}{\cos (w)})}^{2}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.4

Aplique a regra do produto a $\frac{\sin (w)}{\cos (w)}$ .

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \frac{\sin^{2} (w)}{\cos^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.5

Cancele o fator comum de $\cos^{2} (w)$ .

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Etapa 2.1.1.5.1

Cancele o fator comum.

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \frac{\sin^{2} (w)}{\cos^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.5.2

Reescreva a expressão.

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \sin^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.6

Mova $\sin^{2} (w)$ .

$\frac{- 1 + \sin^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.7

Reescreva $- 1$ como $- 1 (1)$ .

$\frac{- 1 (1) + \sin^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.8

Fatore $- 1$ de $\sin^{2} (w)$ .

$\frac{- 1 (1) - 1 (- \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.9

Fatore $- 1$ de $- 1 (1) - 1 (- \sin^{2} (w))$ .

$\frac{- 1 (1 - \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.10

Reescreva $- 1 (1 - \sin^{2} (w))$ como $- (1 - \sin^{2} (w))$ .

$\frac{- (1 - \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.11

Aplique a identidade trigonométrica fundamental.

$\frac{- \cos^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.12

Fatore $\cos^{2} (w)$ de $- \cos^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}$ .

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Etapa 2.1.1.12.1

Fatore $\cos^{2} (w)$ de $- \cos^{2} (w)$ .

$\frac{\cos^{2} (w) \cdot - 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.12.2

Fatore $\cos^{2} (w)$ de $\frac{\cos^{2} (w)}{\sin^{2} (w)}$ .

$\frac{\cos^{2} (w) \cdot - 1 + \cos^{2} (w) \frac{1}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.12.3

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

$\frac{\cos^{2} (w) (- 1 + \frac{1}{\sin^{2} (w)})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.13

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

$\frac{\cos^{2} (w) (- 1 + \frac{1^{2}}{\sin^{2} (w)})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.14

Reescreva $\frac{1^{2}}{\sin^{2} (w)}$ como ${(\frac{1}{\sin (w)})}^{2}$ .

$\frac{\cos^{2} (w) (- 1 + {(\frac{1}{\sin (w)})}^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.15

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

$\frac{\cos^{2} (w) (- 1^{2} + {(\frac{1}{\sin (w)})}^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.16

Reordene $- 1^{2}$ e ${(\frac{1}{\sin (w)})}^{2}$ .

$\frac{\cos^{2} (w) ({(\frac{1}{\sin (w)})}^{2} - 1^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.1.17

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados, $a^{2} - b^{2} = (a + b) (a - b)$ em que $a = \frac{1}{\sin (w)}$ e $b = 1$ .

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Etapa 2.1.2

Simplifique o denominador.

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Etapa 2.1.2.1

Reescreva $\csc (w)$ em termos de senos e cossenos.

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{{(\frac{1}{\sin (w)})}^{2}} - \cos^{4} (w) = 0$

Etapa 2.1.2.2

Aplique a regra do produto a $\frac{1}{\sin (w)}$ .

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\frac{1^{2}}{\sin^{2} (w)}} - \cos^{4} (w) = 0$

Etapa 2.1.2.3

Um elevado a qualquer potência é um.

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\frac{1}{\sin^{2} (w)}} - \cos^{4} (w) = 0$

Etapa 2.1.3

Multiplique o numerador pelo inverso do denominador.

$\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.4

Aplique a propriedade distributiva.

$(\cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.5

Combine $\cos^{2} (w)$ e $\frac{1}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \cdot 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.6

Multiplique $\cos^{2} (w)$ por $1$ .

$(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w)) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.7

Expanda $(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w)) (\frac{1}{\sin (w)} - 1)$ usando o método FOIL.

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Etapa 2.1.7.1

Aplique a propriedade distributiva.

$(\frac{\cos^{2} (w)}{\sin (w)} (\frac{1}{\sin (w)} - 1) + \cos^{2} (w) (\frac{1}{\sin (w)} - 1)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.7.2

Aplique a propriedade distributiva.

$(\frac{\cos^{2} (w)}{\sin (w)} \cdot \frac{1}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) (\frac{1}{\sin (w)} - 1)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.7.3

Aplique a propriedade distributiva.

$(\frac{\cos^{2} (w)}{\sin (w)} \cdot \frac{1}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8

Simplifique e combine termos semelhantes.

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Etapa 2.1.8.1

Simplifique cada termo.

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Etapa 2.1.8.1.1

Combine.

$(\frac{\cos^{2} (w) \cdot 1}{\sin (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.2

Multiplique $\cos^{2} (w)$ por $1$ .

$(\frac{\cos^{2} (w)}{\sin (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.3

Simplifique o denominador.

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Etapa 2.1.8.1.3.1

Eleve $\sin (w)$ à potência de $1$ .

$(\frac{\cos^{2} (w)}{\sin^{1} (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.3.2

Eleve $\sin (w)$ à potência de $1$ .

$(\frac{\cos^{2} (w)}{\sin^{1} (w) \sin^{1} (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.3.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$(\frac{\cos^{2} (w)}{{\sin (w)}^{1 + 1}} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.3.4

Some $1$ e $1$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.4

Combine $\frac{\cos^{2} (w)}{\sin (w)}$ e $- 1$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{\cos^{2} (w) \cdot - 1}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.5

Mova $- 1$ para a esquerda de $\cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{- 1 \cdot \cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.6

Mova o número negativo para a frente da fração.

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.7

Combine $\cos^{2} (w)$ e $\frac{1}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.8

Mova $- 1$ para a esquerda de $\cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} - 1 \cdot \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.1.9

Reescreva $- 1 \cos^{2} (w)$ como $- \cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.2

Some $- \frac{\cos^{2} (w)}{\sin (w)}$ e $\frac{\cos^{2} (w)}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + 0 - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.8.3

Some $\frac{\cos^{2} (w)}{\sin^{2} (w)}$ e $0$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.9

Aplique a propriedade distributiva.

$\frac{\cos^{2} (w)}{\sin^{2} (w)} \sin^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.10

Cancele o fator comum de $\sin^{2} (w)$ .

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Etapa 2.1.10.1

Cancele o fator comum.

$\frac{\cos^{2} (w)}{\sin^{2} (w)} \sin^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.10.2

Reescreva a expressão.

$\cos^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.11

Multiplique por $1$ .

$\cos^{2} (w) \cdot 1 - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.12

Fatore $\cos^{2} (w)$ de $- \cos^{2} (w) \sin^{2} (w)$ .

$\cos^{2} (w) \cdot 1 + \cos^{2} (w) (- 1 \sin^{2} (w)) - \cos^{4} (w) = 0$

Etapa 2.1.13

Fatore $\cos^{2} (w)$ de $\cos^{2} (w) \cdot 1 + \cos^{2} (w) (- 1 \sin^{2} (w))$ .

$\cos^{2} (w) \cdot (1 - 1 \sin^{2} (w)) - \cos^{4} (w) = 0$

Etapa 2.1.14

Reescreva $- 1 \sin^{2} (w)$ como $- \sin^{2} (w)$ .

$\cos^{2} (w) \cdot (1 - \sin^{2} (w)) - \cos^{4} (w) = 0$

Etapa 2.1.15

Aplique a identidade trigonométrica fundamental.

$\cos^{2} (w) \cdot \cos^{2} (w) - \cos^{4} (w) = 0$

Etapa 2.1.16

Multiplique $\cos^{2} (w)$ por $\cos^{2} (w)$ somando os expoentes.

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Etapa 2.1.16.1

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

${\cos (w)}^{2 + 2} - \cos^{4} (w) = 0$

Etapa 2.1.16.2

Some $2$ e $2$ .

$\cos^{4} (w) - \cos^{4} (w) = 0$

Etapa 2.2

Subtraia $\cos^{4} (w)$ de $\cos^{4} (w)$ .

$0 = 0$

Etapa 3

O domínio da expressão consiste em todos os números reais, exceto quando a expressão é indefinida. Nesse caso, não existe um número real que torne a expressão indefinida.