erweitern (e^tsin(t)-e^tcos(t))^2

Lösung

erweitern

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

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

Schritte zur Lösung

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

Wende Formel für perfekte quadratische Gleichungen an:

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

a = e^{t} sin (t), b = e^{t} cos (t)

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

Vereinfache

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

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

Verwende die Doppelwinkelidentität:

2 sin (x) cos (x) = sin (2 x)

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

Verwende die Pythagoreische Identität:

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

= e^{2 t} (- sin (2 t) + 1)

e x p an d \frac{2 n + 1}{2 ( n + 1 )}

e x p an d \frac{11}{( 2 s + 5 ) ^{4}}

e x p an d (\frac{1}{2} + \frac{2 a ^{2}}{b})^{2}

e x p an d \frac{1}{( 2 + j w ) ^{2} ( 4 + j w )}

e x p an d (d + e)^{3}