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beweisen 1+cot^2(x)=(tan^2(x)+1)/(tan^2(x))

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

1 + cot^{2} (x)

Drücke mit sin, cos aus

1 + cot^{2} (x)

Verwende die grundlegende trigonometrische Identität:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

Vereinfache

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

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

Wende Exponentenregel an:

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

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

Wandle das Element in einen Bruch um:

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

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

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

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

Multipliziere:

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

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

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

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

Manipuliere die rechte Seite

\frac{tan ^{2} ( x ) + 1}{tan ^{2} ( x )}

Drücke mit sin, cos aus

\frac{1 + tan ^{2} ( x )}{tan ^{2} ( x )}

Verwende die grundlegende trigonometrische Identität:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

Vereinfache

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

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

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

Wende Exponentenregel an:

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

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

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}

Wende Exponentenregel an:

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

= \frac{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}

Wende Bruchregel an:

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

= \frac{( 1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} ) cos ^{2} ( x )}{sin ^{2} ( x )}

Füge

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

zusammen:

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

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

Wandle das Element in einen Bruch um:

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

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

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

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

Multipliziere:

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

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

= \frac{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ^{2} ( x )} cos ^{2} ( x )}{sin ^{2} ( x )}

Multipliziere

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

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

Multipliziere Brüche:

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

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

Streiche die gemeinsamen Faktoren:

cos^{2} (x)

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

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

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

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

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

\Rightarrow Wahr

p ro v e sin (18 0^{\circ} + θ) = - sin (θ)

p ro v e \frac{1 - tan ^{2} ( x )}{1 + tan ^{2} ( x )} = sec (x)

p ro v e 2 sec (x) = 2 + 2 tan (x)

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

p ro v e sin (x) tan (x) = cos (x)