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4. Czy poniższa zależność jest tożsamością trygonometryczna? a) [latex]1 + sinalpha =left(sinfrac{alpha}{2}+cosfrac{alpha}{2} ight) ^2[/latex] [latex]P=left(sinfrac{alpha}{2}+cosfrac{alpha}{2} ight) ^2=[/latex] [latex]sin^2frac{alpha}{2}+2sinfrac{alpha}{2}cosfrac{alpha}{2}+cos^2frac{alpha}{2}=[/latex] [latex]1+2sinfrac{alpha}{2}cosfrac{alpha}{2}=[/latex] [latex]1+sinleft(2 cdot frac{alpha}{2} ight) =[/latex] [latex]1+sinalpha=L[/latex] c) [latex]sin^4alpha+cos^4alpha=1-frac{sin^22alpha}{2}[/latex] [latex]P=1-frac{sin^22alpha}{2}=[/latex] [latex]1-frac{(2sinalpha cosalpha)^2}{2}=[/latex] [latex]1-frac{4sin^2alpha cos^2alpha}{2}=[/latex] [latex]1-frac{4sin^2alpha cos^2alpha}{2}=[/latex] [latex]1-2sin^2alpha cos^2alpha=[/latex] [latex]sin^2alpha+cos^2alpha -sin^2alpha cos^2alpha-sin^2alpha cos^2alpha=[/latex] [latex]sin^2alpha(1-cos^2alpha) +cos^2alpha(1-sin^2alpha)=[/latex] [latex]sin^2alpha sin^2alpha +cos^2alpha cos^2alpha)=[/latex] [latex]sin^4alpha +cos^4alpha =P[/latex] 5. Udowodnij tożsamość trygonometryczną. b) [latex]cos2alpha=frac{1-tg^2alpha}{1+tg^2alpha}[/latex] [latex]P=frac{1-tg^2alpha}{1+tg^2alpha}=[/latex] [latex]frac{1- frac{sin^2alpha}{cos^2alpha} }{1+ frac{sin^2alpha}{cos^2alpha}}=[/latex] [latex]frac{ frac{cos^2alpha-sin^2alpha}{cos^2alpha} }{frac{cos^2alpha+sin^2alpha}{cos^2alpha}}=[/latex] [latex] frac{cos^2alpha-sin^2alpha}{cos^2alpha+sin^2alpha}=[/latex] [latex] frac{cos2alpha}{1}=cos2alpha=L[/latex] d) [latex]ctgalpha-ctgeta=frac{sin(eta-alpha)}{sinalpha sineta}[/latex] [latex]L=ctgalpha-ctgeta=[/latex] [latex]frac{cosalpha}{sinalpha} - frac{coseta}{sineta} =[/latex] [latex]frac{cosalpha sineta -coseta sinalpha}{sinalpha sineta} =[/latex] [latex]frac{sin(eta-alpha)}{sinalpha sineta}=P[/latex] 6. Uprość wyrażenie. c) [latex]cos(270^o + 2alpha)+sin(450^o+2alpha)=[/latex] [latex]-sin 2alpha+sin(360^o+90^o+2alpha)=[/latex] [latex]-sin 2alpha+sin(90^o+2alpha)=[/latex] [latex]-sin 2alpha+cos 2alpha=[/latex] [latex]cos 2alpha-sin 2alpha[/latex] f) [latex]frac{ sin^2(frac{pi}{2}+alpha)-cos^2(alpha-frac{pi }{2})}{tg^2(frac{pi}{2}+alpha)-ctg^2(alpha-frac{pi }{2})}=[/latex] [latex]frac{ cos^2alpha-cos^2left[-(frac{pi }{2}-alpha) ight] }{ctg^2alpha-ctg^2left[-(frac{pi }{2}-alpha) ight] }=[/latex] [latex]frac{ cos^2alpha-cos^2(frac{pi}{2} -alpha)}{ctg^2alpha-ctg^2(frac{pi }{2}-alpha)}=[/latex] [latex]frac{ cos^2alpha-sin^2alpha}{ctg^2alpha-tg^2alpha}=[/latex] [latex]frac{ cos^2alpha-sin^2alpha}{ frac{cos^2alpha}{sin^2alpha} - frac{sin^2alpha}{cos^2alpha} }=[/latex] [latex]frac{ cos^2alpha-sin^2alpha}{ frac{cos^4alpha-sin^4alpha}{sin^2alpha cos^2alpha} }=[/latex] [latex]frac{ cos^2alpha-sin^2alpha}{ frac{(cos^2alpha-sin^2alpha)(cos^2alpha+sin^2alpha)}{sin^2alpha cos^2alpha} }=[/latex] [latex]sin^2alpha cos^2alpha[/latex]

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