Question

3 ctg^2x = 2tg^x - 5

Answer 1

3ctg²x-2 tg²x+5=0 ,  ctgx=1/tgx

 $frac {3}{tg^{2}x} -2tg^{2}x+5=0 , \ tgx=t \ frac {3}{t^{2}} -2t^{2} +5=0 , \ 3-2t^{4} +5t^{2} =0 , tneq 0 \t^{2}=y \2y^{2} -5y-3=0 \D=25+24=49 , \y_1 =-frac {-1}{2} , y_2=3 \t^{2}neq-frac{1}{2} , t^{2}=3 \ t_1=-sqrt{3} , t_2=sqrt{3} \tgx=-sqrt{3} , x=arctg(-sqrt{3})+pi n , x=-frac{pi}{3}+pi n \tgx-sqrt{3} , x=arctgsqrt{3}+pi k, x=frac{pi}{3}+pi k$ 

n∈Z , k∈Z.