Zadanie 1. Oblicz wartości pozostałych funkcji trygonometrycznych kąta α, gdy cos α =⅔ (dwie trzecie)   Zadanie 2. Uzasadniej tożsamość: ctg α / tg α+ctgα=(1-sin α)(1+sin α) / - kreska ułamkowa α - alfa ctg - cotangens tg - tangens sin - sinus

Zadanie 1.   [latex]cosalpha=frac23\sin^2alpha+cos^2alpha=1\sinalpha=sqrt{1-cos^2alpha}\sinalpha=sqrt{1-(frac23)^2}\sinalpha=sqrt{1-frac49}\sinalpha=sqrt{frac59}\sinalpha=frac{sqrt5}{3}vee sinalpha=-frac{sqrt5}{3}[/latex]   [latex]tgalpha=frac{sinalpha}{cosalpha}\tgalpha=frac{sqrt5}{3} : frac23=frac{sqrt5}{3}cdotfrac32=frac{sqrt5}{2}vee tgalpha=-frac{sqrt5}{2}[/latex]   [latex]ctgalpha=frac{cosalpha}{sinalpha}\ctgalpha=frac{2}{sqrt5}cdotfrac{sqrt5}{sqrt5}=frac{2sqrt5}{5}vee ctgalpha=-frac{2sqrt5}{5}[/latex]   Zadannie 2.   [latex]frac{ctgalpha}{tgalpha+ctgalpha}=(1-sinalpha)(1+sinalpha)[/latex]   [latex]L=frac{ctgalpha}{tgalpha+ctgalpha}=frac{frac{cosalpha}{sinalpha}}{frac{sinalpha}{cosalpha}+frac{cosalpha}{sinalpha}}=[/latex] [latex]=frac{cosalpha}{sinalpha} : frac{sinalphacdot sinalpha+cosalphacdot cosalpha}{cosalphacdot sinalpha}=[/latex] [latex]=frac{cosalpha}{sinalpha} : frac{sin^2alpha+cos^2alpha}{sinalphacdot cosalpha}=frac{cosalpha}{sinalpha} : frac{1}{sinalpha cosalpha}=[/latex] [latex]=frac{cosalpha}{sinalpha}cdotfrac{sinalpha cosalpha}{1}=cos^2alpha=1-sin^2alpha=[/latex] [latex]=(1-sinalpha)(1+sinalpha)=P[/latex]