Question

3×cosx/4×cosx/2×sinx/4 =1-ctgx/1-ctg²x

Answer 1

$3cdot cos frac{x}{4}cdot cos frac{x}{2} cdot sin frac{x}{4}= frac{1-ctgx}{1-ctg^2x}$

$star ; ; 3cdot cos frac{x}{4} cdot cos frac{x}{2} cdot sin frac{x}{4}= 3cdot ( underbrace {sinfrac{x}{4} cdot cos frac{x}{4}}_{1/2cdot sin(2cdot x/4)})cdot cos frac{x}{2}=\\=frac{3}{2}cdot sin(frac{2x}{4})cdot cos frac{x}{2}= frac{3}{2}cdot sin frac{x}{2}cdot cos frac{x}{2}= frac{3}{4}cdot frac{1}{2} cdot sinfrac{2x}{2} =frac{3}{8}cdot sinx; ;star \\\star ; ; frac{1-ctgx}{1-ctg^2x} = frac{1-ctgx}{(1-ctgx)(1+ctgx)} = frac{1}{1+ctgx}= frac{1}{1+frac{cosx}{sinx}} =frac{sinx}{sinx+cosx}; ; star$


$frac{3}{8}cdot sinx= frac{sinx}{sinx+cosx}\\ frac{3}{8}cdot sinx(sinx+cosx)=sinx\\ 3sinx(sinx+cosx)-8sinx=0\\sinxcdot (3sinx+cosx-8)=0\\a); ; sinx=0; ,; ; ; x=pi n,; nin Z\\b); ; 3sinx+cosx=8; |:sqrt{10}\\frac{3}{sqrt{10}}cdot sinx+frac{1}{sqrt{10}}cdot cosx=frac{8}{sqrt{10}}$

$( frac{3}{sqrt{10}})^2+( frac{1}{sqrt{10}})^2=1; ; to ; ; ; frac{3}{sqrt{10}}=sinvarphi ; ,; ; frac{1}{sqrt{10}} =cosvarphi \\tgvarphi =3; ; to ; ; varphi =arctg3\\sinvarphi cdot sinx+cosvarphi cdot cosx=frac{8}{sqrt{10}}$

$cos(x-varphi )=frac{8}{sqrt{10}}; ,; ; ; frac{8}{sqrt{10}}approx frac{8}{3,16}approx 2,53 textgreater 1; ; to \\net; ; teshenij; ,; t.k.; ; |cos alpha | leq 1\\Otvet:; ; x=pi n,; nin Z; .$