find all solutions of the equation in the Interval (0, 2pi)?

Question

sec 4x = 2

tan 3x = 1

Answer 1

sec 4x = 2
4x =sec-1 2=pi/3 or 5pi/3
x=pi/12 or 5pi/12

tan 3x = 1
3x= pi/4 and 5pi/4
x=pi/12 and 5pi/12

Answer 2

The first answer is not right.

If x is to be in (0,2pi), then 4x is to be in (0,8pi).

Since there are two answers with 4x in (0,2pi), there are 4x2=8 answers with 4x in (0,8pi)

sec = 2 when cos = 1/2, i.e. when 4x = pi/3 or 2pi - pi/3 = 5pi/3 or either of things plus 2pi or either plus 4pi or either plus 6pi. Divide each of those eight possibilities by 4 to get the eight possible answers for x.

Same idea with the tan. There are 6 possible values in (0,6pi) for 3x, and hence 6 possible values for x. The first two answers are 3x = pi/2 and 3x = pi + pi/2