Precalculus, Chapter 5, 5.3, Section 5.3, Problem 38

sec(x)+tan(x)=1
1/cos(x)+sin(x)/cos(x)=1
1+sin(x)=cos(x)
1+sin(x)-cos(x)=0
1+cos(pi/2-x)-cos(x)=0
1+(2sin((pi/2-x+x)/2)sin((x-pi/2+x)/2)=0
1+2sin(pi/4)sin(x-pi/4)=0
1+2(1/sqrt(2))(cos(pi/2-x+pi/4)=0
1+sqrt(2)(cos((3pi)/4-x)=0
1+sqrt(2)(-cos(pi-((3pi)/4-x))=0
1-sqrt(2)cos(pi/4+x)=0
cos(pi/4+x)=1/sqrt(2)
General solutions for cos(pi/4+x)=1/ are,
pi/4+x=pi/4+2pin , (7pi)/4+2pin
solving above,
x=2pin , x=(6pi)/4+2pin
solutions in the range 0<=x<=2pi are,
x=0 ,2pi , (3pi)/2
since the equation is undefined for x=3pi/2,
Therefore solutions are x=0 and 2pi