Question

solve each on the interval (0, 2pi)

1. tan(x)sec(x)=3tan(x)
2. tan^2(x)-3tan(x)-2=0

Answer 1

1. tan(x)sec(x)=3tan(x)
tan(x)/cos(x)=3tan(x)
a)tan(x)=0 <=> sin(x)=0 => answer on the required interval is x=pi

b) tan(x)<>0
cos(x)=1/3
x==-arccos(1/3)+2pik where k-ineteger

final answer is combined from answers a),b)

2. tan^2(x)-3tan(x)-2=0
Let's denote& tan(x)=t

t^2-3t-2=0
t=(3+sqrt(17))/2,t=(3-sqrt(17))/2
a)tan(x)=(3+sqrt(17))/2
x=arctan((3+sqrt(17))/2)+pi k, k-integer
b) tan(x)=(3-sqrt(17))/2
x=arctan((3-sqrt(17))/2)+pi n , n-integerfinal answer is combined from answers a),b)