English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

beweisen csc^2(x)-cos^2(x)csc^2(x)=1

Lösung

beweisen

csc^{2} (x) - cos^{2} (x) csc^{2} (x) = 1

Wahr

Schritte zur Lösung

csc^{2} (x) - cos^{2} (x) csc^{2} (x) = 1

Manipuliere die linke Seite

csc^{2} (x) - cos^{2} (x) csc^{2} (x)

Drücke mit sin, cos aus

csc^{2} (x) - cos^{2} (x) csc^{2} (x)

Verwende die grundlegende trigonometrische Identität:

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

= (\frac{1}{sin ( x )})^{2} - cos^{2} (x) (\frac{1}{sin ( x )})^{2}

Vereinfache

(\frac{1}{sin ( x )})^{2} - cos^{2} (x) (\frac{1}{sin ( x )})^{2} : \frac{1 - cos ^{2} ( x )}{sin ^{2} ( x )}

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

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( x )}

Wende Regel an

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

cos^{2} (x) (\frac{1}{sin ( x )})^{2} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

cos^{2} (x) (\frac{1}{sin ( x )})^{2}

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( x )}

Wende Regel an

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

= \frac{1}{sin ^{2} ( x )} cos^{2} (x)

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ^{2} ( x )}{sin ^{2} ( x )}

Multipliziere:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1}{sin ^{2} ( x )} - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

Wende Regel an

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 - cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 - cos ^{2} ( x )}{sin ^{2} ( x )}

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x )}{sin ^{2} ( x )}

Wende Regel an

\frac{a}{a} = 1

= 1

= 1

Wir haben gezeigt, dass beide Seiten die gleiche Form annehmen können

\Rightarrow Wahr

p ro v e tan (\frac{π}{2} - u) = cot (u)

p ro v e csc (θ) - cot (θ) = \frac{sin ( θ )}{1 + cos ( θ )}

p ro v e tan (2 π - θ) = - tan (θ)

p ro v e sin^{4} (θ) - cos^{4} (θ) = 1 - 2 cos^{2} (θ)

p ro v e \frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )} = 0