Trigonometrie Beispiele

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

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

Schritt 1

Beginne auf der rechten Seite.

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

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

Schritt 2

Vereinfache den Ausdruck.

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Schritt 2.1

sin (t)

$\sin (t)$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit

\frac{sin (t) + cos (t)}{sin (t) + cos (t)}

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

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

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

Schritt 2.2

Vereinige die Zähler über dem gemeinsamen Nenner.

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

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

Schritt 2.3

Vereinfache den Zähler.

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Schritt 2.3.1

Wende das Distributivgesetz an.

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

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

Schritt 2.3.2

Multipliziere

sin (t) sin (t)

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

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Schritt 2.3.2.1

Potenziere

sin (t)

$\sin (t)$ mit

1

$1$ .

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

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

Schritt 2.3.2.2

Potenziere

sin (t)

$\sin (t)$ mit

1

$1$ .

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

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

Schritt 2.3.2.3

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

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

Schritt 2.3.2.4

Addiere

1

$1$ und

1

$1$ .

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

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

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

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

Schritt 2.3.3

Bewege

- 1

$- 1$ .

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

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

Schritt 2.3.4

Stelle

{sin}^{2} (t)

$\sin^{2} (t)$ und

- 1

$- 1$ um.

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

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

Schritt 2.3.5

Schreibe

- 1

$- 1$ als

- 1 (1)

$- 1 (1)$ um.

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

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

Schritt 2.3.6

Faktorisiere

- 1

$- 1$ aus

{sin}^{2} (t)

$\sin^{2} (t)$ heraus.

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

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

Schritt 2.3.7

Faktorisiere

- 1

$- 1$ aus

- 1 (1) - 1 (- {sin}^{2} (t))

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

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

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

Schritt 2.3.8

Schreibe

- 1 (1 - {sin}^{2} (t))

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

- (1 - {sin}^{2} (t))

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

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

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

Schritt 2.3.9

Wende den trigonometrischen Pythagoras an.

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

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

Schritt 2.3.10

Faktorisiere

cos (t)

$\cos (t)$ aus

- {cos}^{2} (t) + sin (t) cos (t)

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

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Schritt 2.3.10.1

Faktorisiere

cos (t)

$\cos (t)$ aus

- {cos}^{2} (t)

$- \cos^{2} (t)$ heraus.

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

Schritt 2.3.10.2

Faktorisiere

$\cos (t)$ aus

$\sin (t) \cos (t)$ heraus.

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

Schritt 2.3.10.3

Faktorisiere

$\cos (t)$ aus

$\cos (t) (- \cos (t)) + \cos (t) \sin (t)$ heraus.

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

Schritt 2.4

$\cos (t)$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

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

Schritt 2.5

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 2.6

Vereinfache den Zähler.

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Schritt 2.6.1

Faktorisiere

$\cos (t)$ aus

$\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))$ heraus.

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

Schritt 2.6.2

Addiere

$- \cos (t)$ und

$\cos (t)$ .

$\frac{\cos (t) (0 + \sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

Schritt 2.6.3

Addiere

$0$ und

$\sin (t)$ .

$\frac{\cos (t) (\sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

Schritt 2.6.4

Addiere

$\sin (t)$ und

$\sin (t)$ .

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

Schritt 2.7

Bringe

$2$ auf die linke Seite von

$\cos (t)$ .

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

Schritt 3

Stelle die Terme um.

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

Schritt 4

Schreibe

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

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}$ um.

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

Schritt 5

Da gezeigt wurde, dass die beiden Seiten äquivalent sind, ist die Gleichung eine Identitätsgleichung.

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$ ist eine Identitätsgleichung