Trigonometrie Beispiele

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

Schritt 1

Subtrahiere $\frac{\sin (x)}{\sin (x) + \cos (x)}$ von beiden Seiten der Gleichung.

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

Schritt 2

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

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Schritt 2.1

Vereinfache jeden Term.

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Schritt 2.1.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

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

Schritt 2.1.2

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

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

Schritt 2.1.3

Multipliziere den Zähler mit dem Kehrwert des Nenners.

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

Schritt 2.1.4

Mutltipliziere $\frac{\sin (x)}{\cos (x)}$ mit $\frac{1}{1 + \frac{\sin (x)}{\cos (x)}}$ .

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

Schritt 2.2

Um $\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)}$ .

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

Schritt 2.3

Um $- \frac{\sin (x)}{\sin (x) + \cos (x)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ .

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

Schritt 2.4

Schreibe jeden Ausdruck mit einem gemeinsamen Nenner von $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ , indem du jeden mit einem entsprechenden Faktor von $1$ multiplizierst.

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Schritt 2.4.1

Mutltipliziere $\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ mit $\frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)}$ .

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

Schritt 2.4.2

Mutltipliziere $\frac{\sin (x)}{\sin (x) + \cos (x)}$ mit $\frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ .

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

Schritt 2.4.3

Stelle die Faktoren von $(\sin (x) + \cos (x)) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))$ um.

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

Schritt 2.5

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 2.6

Vereinfache den Zähler.

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Schritt 2.6.1

Faktorisiere $\sin (x)$ aus $\sin (x) (\sin (x) + \cos (x)) - \sin (x) \cos (x) (1 + \frac{\sin (x)}{\cos (x)})$ heraus.

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Schritt 2.6.1.1

Faktorisiere $\sin (x)$ aus $- \sin (x) \cos (x) (1 + \frac{\sin (x)}{\cos (x)})$ heraus.

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

Schritt 2.6.1.2

Faktorisiere $\sin (x)$ aus $\sin (x) (\sin (x) + \cos (x)) + \sin (x) (- \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))$ heraus.

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

Schritt 2.6.2

Wende das Distributivgesetz an.

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

Schritt 2.6.3

Mutltipliziere $- 1$ mit $1$ .

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

Schritt 2.6.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 2.6.4.1

Faktorisiere $\cos (x)$ aus $- \cos (x)$ heraus.

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

Schritt 2.6.4.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.6.4.3

Forme den Ausdruck um.

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

Schritt 2.6.5

Subtrahiere $\sin (x)$ von $\sin (x)$ .

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

Schritt 2.6.6

Addiere $0$ und $\cos (x)$ .

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

Schritt 2.6.7

Subtrahiere $\cos (x)$ von $\cos (x)$ .

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

Schritt 2.7

Mutltipliziere $\sin (x)$ mit $0$ .

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

Schritt 2.8

Dividiere $0$ durch $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ .

$0 = 0$

Schritt 3

Der Definitionsbereich umfasst alle reellen Zahlen, ausgenommen jene, für die der Ausdruck nicht definiert ist. In diesem Fall gibt es keine reellen Zahlen, für die der Ausdruck nicht definiert ist.