Question

Find all solutions of the equation: cot x sec x = 2 cot x.

A) x = 2ð/3 + nð or x = 4ð/3 + nð or 0 + nð
B) x = 2ð/3 + 2nð or x = 4ð/3 + 2nð or 0 + nð
C) x = ð/3 + 2nð or x = 5ð/3 + 2nð or 0 + nð
D) x = ð/3 + nð or x = 5ð/3 + nð or 0 + nð

Answer 1

cot x sec x = 2 cot x.
cotx secx - 2cot x = 0
cotx(secx - 2)
cotx = 0 or secx = 2

if cotx = 0
the cotangent is zero when the tangent is undefined, and the tangent is undefined at 90º (pi/2) and 270º (3pi/2)
or
cosx/sinx = 0
cosx = 0 , x = pi/2 or 3pi/2

if secx = 2
cosx = 1/2, x = pi/3 or 5pi/3

add n(2pi), where n is an integer, to get the general solutions.

Answer 2

Who the hell is teaching 2nd graders this!?