Question

Если tg(L+пи/4)=-1/3.Найдите значение ctg2L.Помогите пожалуйста.

Answer 1

$tg(L+frac{pi}{4})=frac{sin(2L+frac{pi}{2})}{cos(2L+frac{pi}{2})+1} = -frac{1}{3}\ 3sin(2L+frac{pi}{2})=-(cos(2L+frac{pi}{2})+1)\ 3cos2L=-(-sin2L+1)\ 3cos2L=sin2L-1\ 3(2cos^2L-1)=2sinL*cosL-1\ 6cos^2L-3=2sqrt{1-cos^2L}*cosL-1\ cosL=t\ 6t^2-3=2sqrt{1-t^2}*t-1\ t=frac{sqrt{2}}{2}\ L=frac{pi}{4}\ \ t=frac{-sqrt{5}}{5}\ cosL=frac{-sqrt{5}}{5}\ sinL=frac{sqrt{20}}{5}}\ ctgL=-frac{1}{2}\ L=arctg(-0.5)\ ctg2L=ctg(2arctg(-0.5))=-frac{3}{4}$