erweitern ((sin(x)cos(x)+1)^2)/(sin^2(x)cos^2(x))

Lösung

erweitern

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

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

Schritte zur Lösung

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

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

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

Wende Bruchregel an:

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

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

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

Wende Regel an

\frac{a}{a} = 1

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

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

e x p an d 6 e^{x} - 3 (e^{x} x^{2} + 2 x e^{x})

e x p an d (0.5 e^{x} + 0.5) (0.5 e^{- x} + 0.5)

e x p an d \frac{3}{( 1 + x ) ( 2 x - 1 )}

e x p an d 2 (- sin (a) cos (3 a) - 3 cos (a) sin (3 a)) + sin (2 a)

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