Help with Trig Equations

Question

csc²(x/2)=2sec(x)

Please help me solve for x.

Answer 1

Take reciprocals,

sin²(x/2) = cos(x)/2

=> 2sin²(x/2) = cos(x)

=> 1-cos(x) = cos(x)

=> cos(x) = 1/2

=> x = pi/3, 5pi/3 on [0, 2pi).