udowodnij tozsamosc

[logs4]

Udowodnij: \(\displaystyle{ \frac{sinx}{1+cosx}+ \frac{1+cosx}{sinx} =2cosecx}\)

[lorakesz]

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

[czeslaw]

\(\displaystyle{ \frac{sinx}{1+cosx}+ \frac{1+cosx}{sinx} = \frac{sin^{2}x+(1+cosx)^{2}}{sinx(1+cosx)} = \frac{sin^{2}x+1+2cosx+cos^{2}x}{sinx(1+cosx)} = \frac{2(1+cosx)}{sinx(1+cosx)} = \frac{2}{sinx}=2cosecx}\)