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beweisen (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5

Lösung

beweisen

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

Wahr

Schritte zur Lösung

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

Manipuliere die linke Seite

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

Faktorisiere

(- sin (x) + 2 cos (x))^{2} + (cos (x) + 2 sin (x))^{2} : 5 (sin^{2} (x) + cos^{2} (x))

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

(- sin (x) + 2 cos (x))^{2} : sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x)

Wende Formel für perfekte quadratische Gleichungen an:

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

a = - sin (x), b = 2 cos (x)

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

Vereinfache

(- sin (x))^{2} + 2 (- sin (x)) \cdot 2 cos (x) + (2 cos (x))^{2} : sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x)

(- sin (x))^{2} + 2 (- sin (x)) \cdot 2 cos (x) + (2 cos (x))^{2}

Entferne die Klammern:

(- a) = - a

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

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

(- sin (x))^{2}

Wende Exponentenregel an:

(- a)^{n} = a^{n},

wenn

n

gerade ist

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

= sin^{2} (x)

2 sin (x) \cdot 2 cos (x) = 4 sin (x) cos (x)

2 sin (x) \cdot 2 cos (x)

Multipliziere die Zahlen:

2 \cdot 2 = 4

= 4 sin (x) cos (x)

(2 cos (x))^{2} = 4 cos^{2} (x)

(2 cos (x))^{2}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

= 4 cos^{2} (x)

= sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x)

= sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x)

= sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x) + (cos (x) + 2 sin (x))^{2}

(cos (x) + 2 sin (x))^{2} : cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

Wende Formel für perfekte quadratische Gleichungen an:

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

a = cos (x), b = 2 sin (x)

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

Vereinfache

cos^{2} (x) + 2 cos (x) \cdot 2 sin (x) + (2 sin (x))^{2} : cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

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

2 cos (x) \cdot 2 sin (x) = 4 cos (x) sin (x)

2 cos (x) \cdot 2 sin (x)

Multipliziere die Zahlen:

2 \cdot 2 = 4

= 4 cos (x) sin (x)

(2 sin (x))^{2} = 4 sin^{2} (x)

(2 sin (x))^{2}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

= 4 sin^{2} (x)

= cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

= cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

= sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x) + cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

Vereinfache

sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x) + cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x) : 5 sin^{2} (x) + 5 cos^{2} (x)

sin^{2} (x) - 4 sin (x) cos (x) + 4 cos^{2} (x) + cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

Addiere gleiche Elemente:

- 4 sin (x) cos (x) + 4 cos (x) sin (x) = 0

= sin^{2} (x) + 4 cos^{2} (x) + cos^{2} (x) + 4 sin^{2} (x)

Addiere gleiche Elemente:

4 cos^{2} (x) + cos^{2} (x) = 5 cos^{2} (x)

= sin^{2} (x) + 5 cos^{2} (x) + 4 sin^{2} (x)

Addiere gleiche Elemente:

sin^{2} (x) + 4 sin^{2} (x) = 5 sin^{2} (x)

= 5 sin^{2} (x) + 5 cos^{2} (x)

= 5 sin^{2} (x) + 5 cos^{2} (x)

Klammere gleiche Terme aus

5

= 5 (sin^{2} (x) + cos^{2} (x))

= (cos^{2} (x) + sin^{2} (x)) \cdot 5

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(cos^{2} (x) + sin^{2} (x)) \cdot 5

Verwende die Pythagoreische Identität:

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

= 1 \cdot 5

Vereinfache

= 5

= 5

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

\Rightarrow Wahr

p ro v e cos (a) (sec (a) - cos (a)) = sin^{2} (a)

p ro v e tan (x) cos (x) csc (x) = 1

p ro v e \frac{csc ^{4} ( x ) - 1}{1 + csc ^{2} ( x )} = cot^{2} (x)

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

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