Terme vereinfachen(Sinus,Kosinus,Tangens)

Question

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15 Vereinfache:
a) \( \tan (\alpha) \cdot \cos (\alpha) \)
b) \( \frac{\sin (\alpha)}{\tan (\alpha)} \)
c) \( \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha) \)
d) \( \frac{1}{\tan (\alpha) \cdot \cos (\alpha)} \)
e) \( \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)} \) f) \( \sin (\alpha)+\frac{\cos (\alpha)}{\tan (\alpha)} \)
g) \( \sin ^{4}(\alpha)-\cos ^{4}(\alpha) \)
h) \( \frac{\tan (\alpha)}{\sin (\alpha)}-\tan (\alpha) \cdot \sin (\alpha) \)
i) \( \frac{\cos (\alpha)}{1-\sin (\alpha)}-\frac{1}{\cos (\alpha)} \)

Answer 1

Hallo,

Aufgabe c)

\( \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha) \)

= sin(α) (sin^2(α) +cos^2(α))  --------->sin^2(α) +cos^2(α) =1

=sin(α)

Answer 2

a)  \( \tan (\alpha) \cdot \cos (\alpha) \) =\( \frac{sin(α)}{cos(α)} \)* cos(α) =sin(α)

b)  \( \frac{\sin (\alpha)}{\tan (\alpha)} \) = cos(α)

Answer 3

g)

\( \sin ^{4}(\alpha)-\cos ^{4}(\alpha) \)

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

\( =\sin ^{2}(\alpha)-\cos ^{2}(\alpha)\)

\( =1-\cos ^{2}(\alpha)-\cos ^{2}(\alpha)\)

\( =1-2\cos ^{2}(\alpha)\)

e)

 \( \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)} \)

 \( =\sqrt{(1+\cos (\alpha))(1-\cos (\alpha))} \)

 \( =\sqrt{1^2-\cos^2 (\alpha)} \)

\(=\sin\alpha\)

Für weitere Hilfen solltest du eigene Ansätze posten.

:-)