Trigonometria Esempi

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$

Passaggio 1

Inizia dal lato sinistro.

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)}$

Passaggio 2

Converti in seni e coseni.

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Passaggio 2.1

Scrivi $\tan (a)$ in seno e coseno utilizzando l'identità quoziente.

$\frac{\frac{\sin (a)}{\cos (a)} - \tan (b)}{\cot (b) - \cot (a)}$

Passaggio 2.2

Scrivi $\tan (b)$ in seno e coseno utilizzando l'identità quoziente.

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\cot (b) - \cot (a)}$

Passaggio 2.3

Scrivi $\cot (b)$ in seno e coseno utilizzando l'identità quoziente.

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \cot (a)}$

Passaggio 2.4

Scrivi $\cot (a)$ in seno e coseno utilizzando l'identità quoziente.

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$

Passaggio 3

Semplifica.

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Passaggio 3.1

Multiply the numerator and denominator of the fraction by $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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Passaggio 3.1.1

Moltiplica $\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$ per $\frac{\cos (a) \cos (b) \sin (b) \sin (a)}{\cos (a) \cos (b) \sin (b) \sin (a)}$ .

$\frac{\cos (a) \cos (b) \sin (b) \sin (a)}{\cos (a) \cos (b) \sin (b) \sin (a)} \cdot \frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$

Passaggio 3.1.2

Combina.

$\frac{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)})}$

Passaggio 3.2

Applica la proprietà distributiva.

$\frac{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\sin (a)}{\cos (a)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Passaggio 3.3

Semplifica cancellando.

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Passaggio 3.3.1

Elimina il fattore comune di $\cos (a)$ .

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Passaggio 3.3.1.1

Scomponi $\cos (a)$ da $\cos (a) \cos (b) \sin (b) \sin (a)$ .

$\frac{\cos (a) (\cos (b) \sin (b) \sin (a)) \frac{\sin (a)}{\cos (a)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Passaggio 3.3.1.2

Elimina il fattore comune.

Passaggio 3.3.1.3

Riscrivi l'espressione.

$\frac{\cos (b) \sin (b) \sin (a) \sin (a) + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) \frac{\cos (b)}{\sin (b)} + \cos (a) \cos (b) \sin (b) \sin (a) (- \frac{\cos (a)}{\sin (a)})}$

Passaggio 3.3.2

Eleva $\sin (a)$ alla potenza di $1$ .

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

Passaggio 3.3.3

Eleva $\sin (a)$ alla potenza di $1$ .

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

Passaggio 3.3.4

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 3.3.5

Somma $1$ e $1$ .

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

Passaggio 3.3.6

Elimina il fattore comune di $\cos (b)$ .

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Passaggio 3.3.6.1

Sposta il negativo all'inizio di $- \frac{\sin (b)}{\cos (b)}$ nel numeratore.

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

Passaggio 3.3.6.2

Scomponi $\cos (b)$ da $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Passaggio 3.3.6.3

Elimina il fattore comune.

Passaggio 3.3.6.4

Riscrivi l'espressione.

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

Passaggio 3.3.7

Eleva $\sin (b)$ alla potenza di $1$ .

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

Passaggio 3.3.8

Eleva $\sin (b)$ alla potenza di $1$ .

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

Passaggio 3.3.9

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 3.3.10

Somma $1$ e $1$ .

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

Passaggio 3.3.11

Elimina il fattore comune di $\sin (b)$ .

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Passaggio 3.3.11.1

Scomponi $\sin (b)$ da $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Passaggio 3.3.11.2

Elimina il fattore comune.

Passaggio 3.3.11.3

Riscrivi l'espressione.

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

Passaggio 3.3.12

Eleva $\cos (b)$ alla potenza di $1$ .

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

Passaggio 3.3.13

Eleva $\cos (b)$ alla potenza di $1$ .

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

Passaggio 3.3.14

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 3.3.15

Somma $1$ e $1$ .

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

Passaggio 3.3.16

Elimina il fattore comune di $\sin (a)$ .

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Passaggio 3.3.16.1

Sposta il negativo all'inizio di $- \frac{\cos (a)}{\sin (a)}$ nel numeratore.

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

Passaggio 3.3.16.2

Scomponi $\sin (a)$ da $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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

Passaggio 3.3.16.3

Elimina il fattore comune.

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

Passaggio 3.3.16.4

Riscrivi l'espressione.

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

Passaggio 3.3.17

Eleva $\cos (a)$ alla potenza di $1$ .

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

Passaggio 3.3.18

Eleva $\cos (a)$ alla potenza di $1$ .

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

Passaggio 3.3.19

Utilizza la regola per la potenza di una potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

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

Passaggio 3.3.20

Somma $1$ e $1$ .

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

Passaggio 3.4

Scomponi $\sin (b) \sin (a)$ da $\cos (b) \sin (b) {\sin (a)}^{2} + \cos (a) \sin (a) (- {\sin (b)}^{2})$ .

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Passaggio 3.4.1

Scomponi $\sin (b) \sin (a)$ da $\cos (b) \sin (b) {\sin (a)}^{2}$ .

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

Passaggio 3.4.2

Scomponi $\sin (b) \sin (a)$ da $\cos (a) \sin (a) (- {\sin (b)}^{2})$ .

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

Passaggio 3.4.3

Scomponi $\sin (b) \sin (a)$ da $\sin (b) \sin (a) (\cos (b) \sin (a)) + \sin (b) \sin (a) (\cos (a) (- \sin (b)))$ .

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

Passaggio 3.5

Scomponi $\cos (a) \cos (b)$ da $\cos (a) {\cos (b)}^{2} \sin (a) + \cos (b) \sin (b) (- {\cos (a)}^{2})$ .

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Passaggio 3.5.1

Scomponi $\cos (a) \cos (b)$ da $\cos (a) {\cos (b)}^{2} \sin (a)$ .

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

Passaggio 3.5.2

Scomponi $\cos (a) \cos (b)$ da $\cos (b) \sin (b) (- {\cos (a)}^{2})$ .

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) + \cos (a) (- \sin (b)))}{\cos (a) \cos (b) (\cos (b) \sin (a)) + \cos (a) \cos (b) (\sin (b) (- \cos (a)))}$

Passaggio 3.5.3

Scomponi $\cos (a) \cos (b)$ da $\cos (a) \cos (b) (\cos (b) \sin (a)) + \cos (a) \cos (b) (\sin (b) (- \cos (a)))$ .

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) + \cos (a) (- \sin (b)))}{\cos (a) \cos (b) (\cos (b) \sin (a) + \sin (b) (- \cos (a)))}$

Passaggio 3.6

Riordina i termini.

$\frac{\sin (b) \sin (a) (\cos (b) \sin (a) - 1 \cdot \cos (a) \sin (b))}{\cos (a) \cos (b) (\cos (b) \sin (a) - 1 \cdot \sin (b) \cos (a))}$

Passaggio 3.7

Elimina il fattore comune di $\cos (b) \sin (a) - 1 \cdot \cos (a) \sin (b)$ e $\cos (b) \sin (a) - 1 \cdot \sin (b) \cos (a)$ .

$\frac{\sin (b) \sin (a)}{\cos (a) \cos (b)}$

Passaggio 4

Riordina i termini.

$\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$

Passaggio 5

Riscrivi $\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$ come $\frac{\tan (b)}{\cot (a)}$ .

$\frac{\tan (b)}{\cot (a)}$

Passaggio 6

Poiché si è dimostrato che i due lati sono equivalenti, l'equazione è un'identità.

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$ è un'identità