Elementarmathematik Beispiele

\sec^{2} (x) + \csc^{2} (x) = \sec^{2} (x) \cdot \csc^{2} (x)

$\sec^{2} (x) + \csc^{2} (x) = \sec^{2} (x) \cdot \csc^{2} (x)$

Schritt 1

Subtrahiere

\sec^{2} (x) \cdot \csc^{2} (x)

$\sec^{2} (x) \cdot \csc^{2} (x)$ von beiden Seiten der Gleichung.

\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0

$\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0$

Schritt 2

Ersetze die

\sec^{2} (x)

$\sec^{2} (x)$ durch

1 + \tan^{2} (x)

$1 + \tan^{2} (x)$ basierend auf der

\tan^{2} (x) + 1 = \sec^{2} (x)

$\tan^{2} (x) + 1 = \sec^{2} (x)$ -Identitätsgleichung.

(1 + \tan^{2} (x)) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0

$(1 + \tan^{2} (x)) + \csc^{2} (x) - \sec^{2} (x) \cdot \csc^{2} (x) = 0$

Schritt 3

Mutltipliziere

- \sec^{2} (x)

$- \sec^{2} (x)$ mit

\csc^{2} (x)

$\csc^{2} (x)$ .

1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0

$1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Schritt 4

Vereinfache die linke Seite.

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Schritt 4.1

Vereinfache

1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x)

$1 + \tan^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x)$ .

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Schritt 4.1.1

Ordne Terme um.

\tan^{2} (x) + 1 + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0

$\tan^{2} (x) + 1 + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Schritt 4.1.2

Wende den trigonometrischen Pythagoras an.

\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0

$\sec^{2} (x) + \csc^{2} (x) - \sec^{2} (x) \csc^{2} (x) = 0$

Schritt 4.1.3

Vereinfache durch Herausfaktorisieren.

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Schritt 4.1.3.1

Bewege

- \sec^{2} (x) \csc^{2} (x)

$- \sec^{2} (x) \csc^{2} (x)$ .

\sec^{2} (x) - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0

$\sec^{2} (x) - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0$

Schritt 4.1.3.2

Faktorisiere

- \sec^{2} (x)

$- \sec^{2} (x)$ aus

\sec^{2} (x)

$\sec^{2} (x)$ heraus.

- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) \csc^{2} (x) + \csc^{2} (x) = 0$

Schritt 4.1.3.3

Faktorisiere

- \sec^{2} (x)

$- \sec^{2} (x)$ aus

- \sec^{2} (x) \csc^{2} (x)

$- \sec^{2} (x) \csc^{2} (x)$ heraus.

- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x)) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x)) + \csc^{2} (x) = 0$

Schritt 4.1.3.4

Faktorisiere

- \sec^{2} (x)

$- \sec^{2} (x)$ aus

- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x))

$- \sec^{2} (x) \cdot - 1 - \sec^{2} (x) (\csc^{2} (x))$ heraus.

- \sec^{2} (x) \cdot (- 1 + \csc^{2} (x)) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot (- 1 + \csc^{2} (x)) + \csc^{2} (x) = 0$

- \sec^{2} (x) \cdot (- 1 + \csc^{2} (x)) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot (- 1 + \csc^{2} (x)) + \csc^{2} (x) = 0$

Schritt 4.1.4

Ordne Terme um.

- \sec^{2} (x) \cdot (\csc^{2} (x) - 1) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot (\csc^{2} (x) - 1) + \csc^{2} (x) = 0$

Schritt 4.1.5

Wende den trigonometrischen Pythagoras an.

- \sec^{2} (x) \cdot \cot^{2} (x) + \csc^{2} (x) = 0

$- \sec^{2} (x) \cdot \cot^{2} (x) + \csc^{2} (x) = 0$

Schritt 4.1.6

Vereinfache den Ausdruck.

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Schritt 4.1.6.1

Schreibe

\sec^{2} (x) \cdot \cot^{2} (x)

$\sec^{2} (x) \cdot \cot^{2} (x)$ als

{(\sec (x) \cot (x))}^{2}

${(\sec (x) \cot (x))}^{2}$ um.

- {(\sec (x) \cot (x))}^{2} + \csc^{2} (x) = 0

$- {(\sec (x) \cot (x))}^{2} + \csc^{2} (x) = 0$

Schritt 4.1.6.2

Stelle

- {(\sec (x) \cot (x))}^{2}

$- {(\sec (x) \cot (x))}^{2}$ und

\csc^{2} (x)

$\csc^{2} (x)$ um.

\csc^{2} (x) - {(\sec (x) \cot (x))}^{2} = 0

$\csc^{2} (x) - {(\sec (x) \cot (x))}^{2} = 0$

\csc^{2} (x) - {(\sec (x) \cot (x))}^{2} = 0

$\csc^{2} (x) - {(\sec (x) \cot (x))}^{2} = 0$

Schritt 4.1.7

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ , mit

a = \csc (x)

$a = \csc (x)$ und

b = \sec (x) \cot (x)

$b = \sec (x) \cot (x)$ .

(\csc (x) + \sec (x) \cot (x)) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \sec (x) \cot (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Schritt 4.1.8

Vereinfache Terme.

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Schritt 4.1.8.1

Vereinfache jeden Term.

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Schritt 4.1.8.1.1

Schreibe mithilfe von Sinus und Kosinus um, kürze dann die gemeinsamen Faktoren.

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Schritt 4.1.8.1.1.1

Stelle

\sec (x)

$\sec (x)$ und

\cot (x)

$\cot (x)$ um.

(\csc (x) + \cot (x) \sec (x)) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \cot (x) \sec (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Schritt 4.1.8.1.1.2

Schreibe

\sec (x) \cot (x)

$\sec (x) \cot (x)$ mithilfe von Sinus und Kosinus um.

(\csc (x) + \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0

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

Schritt 4.1.8.1.1.3

Kürze die gemeinsamen Faktoren.

(\csc (x) + \frac{1}{\sin (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \frac{1}{\sin (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0$

(\csc (x) + \frac{1}{\sin (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \frac{1}{\sin (x)}) (\csc (x) - (\sec (x) \cot (x))) = 0$

Schritt 4.1.8.1.2

Wandle von

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ nach

\csc (x)

$\csc (x)$ um.

(\csc (x) + \csc (x)) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \csc (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

(\csc (x) + \csc (x)) (\csc (x) - (\sec (x) \cot (x))) = 0

$(\csc (x) + \csc (x)) (\csc (x) - (\sec (x) \cot (x))) = 0$

Schritt 4.1.8.2

Addiere

\csc (x)

$\csc (x)$ und

\csc (x)

$\csc (x)$ .

2 \csc (x) (\csc (x) - (\sec (x) \cot (x))) = 0

$2 \csc (x) (\csc (x) - (\sec (x) \cot (x))) = 0$

Schritt 4.1.8.3

Vereinfache jeden Term.

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Schritt 4.1.8.3.1

Schreibe mithilfe von Sinus und Kosinus um, kürze dann die gemeinsamen Faktoren.

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Schritt 4.1.8.3.1.1

Stelle

\sec (x)

$\sec (x)$ und

\cot (x)

$\cot (x)$ um.

2 \csc (x) (\csc (x) - (\cot (x) \sec (x))) = 0

$2 \csc (x) (\csc (x) - (\cot (x) \sec (x))) = 0$

Schritt 4.1.8.3.1.2

Schreibe

- (\sec (x) \cot (x))

$- (\sec (x) \cot (x))$ mithilfe von Sinus und Kosinus um.

2 \csc (x) (\csc (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)})) = 0

$2 \csc (x) (\csc (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)})) = 0$

Schritt 4.1.8.3.1.3

Kürze die gemeinsamen Faktoren.

2 \csc (x) (\csc (x) - \frac{1}{\sin (x)}) = 0

$2 \csc (x) (\csc (x) - \frac{1}{\sin (x)}) = 0$

2 \csc (x) (\csc (x) - \frac{1}{\sin (x)}) = 0

$2 \csc (x) (\csc (x) - \frac{1}{\sin (x)}) = 0$

Schritt 4.1.8.3.2

Wandle von

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ nach

\csc (x)

$\csc (x)$ um.

2 \csc (x) (\csc (x) - \csc (x)) = 0

$2 \csc (x) (\csc (x) - \csc (x)) = 0$

2 \csc (x) (\csc (x) - \csc (x)) = 0

$2 \csc (x) (\csc (x) - \csc (x)) = 0$

Schritt 4.1.8.4

Subtrahiere

\csc (x)

$\csc (x)$ von

\csc (x)

$\csc (x)$ .

2 \csc (x) \cdot 0 = 0

$2 \csc (x) \cdot 0 = 0$

2 \csc (x) \cdot 0 = 0

$2 \csc (x) \cdot 0 = 0$

Schritt 4.1.9

Multipliziere

2 \csc (x) \cdot 0

$2 \csc (x) \cdot 0$ .

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Schritt 4.1.9.1

Mutltipliziere

0

$0$ mit

2

$2$ .

0 \csc (x) = 0

$0 \csc (x) = 0$

Schritt 4.1.9.2

Mutltipliziere

0

$0$ mit

\csc (x)

$\csc (x)$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Schritt 5

0 = 0

$0 = 0$ , ist die Gleichung immer erfüllt.

Immer wahr