Find all 0 ≤ x < 2π such that sin x − csc x=0.

Answer 1

sinx = cscx = 1/sinx

sin^2(x) = 1

sinx = + or -1

x = pi/2 or 3pi/2 = 90 or 270 degrees

sin90-csc90 = 1-1 =0

sin270-csc270 =-1 - (-1) =-1+1 = 0

Answer 2

This is equivalent to sin(x) = csc(x) and csc(x) = 1/sin(x), so sin(x) = 1/sin(x) or sin²(x) = 1. This gives sin(x) = ±1, so x = π/2, 3π/2.