Precalculus, Chapter 5, 5.3, Section 5.3, Problem 37

csc(x)+cot(x)=1
1/sin(x)+cos(x)/sin(x)=1
1+cos(x)=sin(x)
1+cos(x)-sin(x)=0
1+cos(x)-cos(pi/2-x)=0
1+2sin((x+pi/2-x)/2)sin((pi/2-x-x)/2)=0
1+2sin(pi/4)sin(pi/4-x)=0
1+2(1/sqrt(2))cos(pi/2-(pi/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/sqrt(2) are
pi/4+x=(3pi)/4+2pin , pi/4+x=(5pi)/4+2pin
solving above,
x=((3pi)/4-pi/4+2pin) , x=((5pi)/4-pi/4+2pin)
x=pi/2+2pin , pi+2pin
solutions for the range 0<=x<=2pi are,
x=pi/2 , pi
since the equation is undefined for pi
Solution is pi/2