a) [latex]1+sin alpha=[/latex] [latex]sin90^o+sinalpha=[/latex] [latex]2sin frac{90^o+alpha}{2}cos frac{90^o-alpha}{2}=[/latex] [latex]2sinleft(45^o+frac{alpha}{2} ight) cosleft(45^o- frac{alpha}{2} ight)[/latex] ====================== b) [latex]frac{1}{2}-cos alpha=[/latex] [latex]cos60^o-cos alpha=[/latex] [latex]-2sin frac{60^o+alpha}{2}sin frac{60^o-alpha}{2}=[/latex] [latex]-2sinleft(30^o+frac{alpha}{2} ight) sinleft(30^o-frac{alpha}{2} ight)[/latex] ====================== c) [latex]1-tg^2 alpha =1- frac{sin^2alpha}{cos^2alpha}= frac{cos^2alpha-sin^2alpha}{cos^2alpha}=[/latex] [latex]frac{cos^2alpha-sin^2alpha}{cos^2alpha}= frac{cos2alpha}{cos^2alpha}=[/latex] [latex]cos2alpha cdot frac{1}{cos^2alpha}[/latex] ====================== d) [latex]sin alpha +sin2 alpha +sin3 alpha=[/latex] [latex](sin alpha +sin3 alpha)+sin2 alpha=[/latex] [latex]2sin frac{alpha+3alpha}{2}cos frac{alpha-3alpha}{2} +sin2 alpha=[/latex] [latex]2sin frac{4alpha}{2}cos frac{-2alpha}{2} +sin2 alpha=[/latex] [latex]2sin 2alpha cos(-alpha)+sin2 alpha=[/latex] [latex]2sin 2alpha cosalpha+sin2 alpha=[/latex] [latex]2sin 2alpha left(cosalpha+ frac{1}{2} ight) =[/latex] [latex]2sin 2alpha left(cosalpha+cos60^o ight)=[/latex] [latex]2sin 2alpha cdot 2cos frac{alpha+60^o}{2}cos frac{alpha-60^o}{2}=[/latex] [latex]4sin 2alpha cosleft(frac{alpha}{2}+30^o ight) cos left(frac{alpha}{2}-30^o ight)[/latex] ====================== e) [latex]cos alpha +cos2 alpha +cos3 alpha=[/latex] [latex](cos alpha +cos3 alpha)+cos2 alpha =[/latex] [latex]2cos frac{alpha+3alpha}{2}cos frac{alpha-3alpha}{2} +cos2 alpha =[/latex] [latex]2cos frac{4alpha}{2}cos frac{-2alpha}{2} +cos2 alpha =[/latex] [latex]2cos2alpha cos(-alpha) +cos2 alpha =[/latex] [latex]2cos2alpha cosalpha +cos2 alpha =[/latex] [latex]2cos2alpha left(cos alpha+frac{1}{2} ight) =[/latex] [latex]2cos2alpha left(cosalpha +cos60^o ight) =[/latex] [latex]2cos2alpha cdot 2cos frac{alpha+60^o}{2}cos frac{alpha-60^o}{2}=[/latex] [latex]4cos2alpha cosleft(frac{alpha}{2}+30^o ight) cos left(frac{alpha}{2}-30^o ight)[/latex] ====================== f) [latex]sin alpha +sin eta +sin(alpha +eta )=[/latex] [latex]2sin frac{alpha+eta}{2}cos frac{alpha-eta}{2} +sinleft(2 cdot frac{alpha+eta}{2} ight) =[/latex] [latex]2sin frac{alpha+eta}{2}cos frac{alpha-eta}{2}+2sinfrac{alpha+eta}{2}cos frac{alpha+eta}{2} =[/latex] [latex]2sin frac{alpha+eta}{2}left(cos frac{alpha-eta}{2}+cos frac{alpha+eta}{2} ight) =[/latex] [latex]2sin frac{alpha+eta}{2} cdot 2cos frac{frac{alpha-eta}{2}+frac{alpha+eta}{2}}{2} cos frac{frac{alpha-eta}{2}-frac{alpha+eta}{2}}{2}=[/latex] [latex]4sin frac{alpha+eta}{2}cos frac{frac{alpha-eta+alpha+eta}{2}}{2} cos frac{frac{alpha-eta-alpha-eta}{2}}{2}=[/latex] [latex]4sin frac{alpha+eta}{2}cos frac{2alpha}{4} cos frac{-2eta}{4}=[/latex] [latex]4sin frac{alpha+eta}{2}cos frac{alpha}{2} cos frac{-eta}{2}=[/latex] [latex]4sin frac{alpha+eta}{2}cos frac{alpha}{2} cos frac{eta}{2}[/latex]
