Question

20 ! 1. cos(a+b)cos(a-b)+sin(a+b)*sin(a-b) 2. tg(a+b)-tgb/1+tg(a+b)tgb 3. cos(a-п/6)-0,5sina

Answer 1

1
cos(a+b)cos(a-b)+sin(a+b)*sin(a-b)=cos(a+b-a+b)=cos2b
2
(tg(a+b)-tgb)/(1+tg(a+b)tgb)=tg(a+b-b)=tga
3
cos(a-п/6)-0,5sina=cosacosπ/6+sinasinπ/6-0,5sina=
=√3/2*cosa+0,5sina-0,5sina=√3/2*cosa

Answer 2

$\mathtt{cos(a+b)cos(a-b)+sin(a+b)sin(a-b)=}\\\mathtt{cos[(a+b)-(a-b)]=cos[a+b-a+b]=cos2b}$

$\mathtt{\frac{tg(a+b)-tgb}{1+tg(a+b)tgb}=tg[a+b-b]=tga}$

$\mathtt{cos(a-\frac{\pi}{6})-\frac{1}{2}sina=cosacos(\frac{\pi}{6})+sinasin(\frac{\pi}{6})-\frac{1}{2}sina=}\\\mathtt{cosa*\frac{\sqrt{3}}{2}+sina*\frac{1}{2}-\frac{1}{2}sina=\frac{\sqrt{3}}{2}cosa}$