tozsamosci trygonometryczne

[Aguskaq]

sprawdz czy podane rownosci sa tozsamosciami trygonometrycznymi :

1. \(\displaystyle{ \frac{1}{1-cos\alpha}}\)+\(\displaystyle{ \frac{1}{1+cos\alpha}=\frac{2}{sin^{2}\alpha}}\)
2.
\(\displaystyle{ \frac{cos\alpha}{1+sin\alpha}}\)+\(\displaystyle{ \frac{cos\alpha}{1-sin\alpha}=\frac{2}{cos\alpha}}\)
3.
\(\displaystyle{ \frac{1+cos\alpha}{sin\alpha}}\)=\(\displaystyle{ \frac{sin\alpha}{1-cos\alpha}}\)
4.
\(\displaystyle{ (\frac{1}{cos\alpha}}\)+\(\displaystyle{ \frac{1}{sin\alpha})*(tg\alpha+ctg\alpha-1)}=\frac{sin\alpha}{cos^{2}\alpha}+\frac{cos\alpha}{sin^{2}\alpha}}\)

[Vixy]

3)mnoze na krzyz

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

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

[Rafal88K]

1. \(\displaystyle{ \frac{1}{1 - cos\alpha} + \frac{1}{1 + cos\alpha} = \frac{1 + cos\alpha + 1 - cos\alpha}{(1 - cos\alpha)(1 + cos\alpha)} = \frac{2}{1 - cos^{2}\alpha} = \frac{2}{sin^{2}\alpha}}\)

L=P

2. \(\displaystyle{ \frac{cos\alpha}{1 + sin\alpha} + \frac{cos\alpha}{1 - sin\alpha} = \frac{cos\alpha - sin\alpha cos\alpha + cos\alpha - sin\alpha cos\alpha}{1 - sin^{2}\alpha} = \frac{2cos\alpha}{cos^{2}\alpha} = \frac{2}{cos\alpha}}\)

L = P

[Vixy]

2) nalezy dac wspolnych mianownik

\(\displaystyle{ \frac{(1-sin\alpha)*cos\alpha+(1+sin\alpha)*cos\alpha}{(1+sin\alpha)(1-sin\alpha)}}\)=\(\displaystyle{ \frac{cos\alpha-sin\alpha*cos\alpha+cos\alpha+sin\alpha*cos\alpha)}{1-sin^\alpha}}\)=\(\displaystyle{ \frac{2co\alpha}{1-(1-cos^\alpha)}}\)=\(\displaystyle{ \frac{2cos\alpha}{cos^2\alpha}}\)=\(\displaystyle{ \frac{2}{cos\alpha}}\)


L=P