Question

How do you solve `secx=tanx+1` for `0<=x<=2pi`?

Answer 1

`0 ; 2pi`

Explanation:

Transform the equation:
`1/cos x = sin x/cos x + 1`
`1/cos x = (sin x + cos x)/cos x`
Simplify the equation, under condition (1) that cos x different to 0, meaning x different to `pi/2 and (3pi)/2`
Next, solve the trig equation
sin x + cos x = 1
Use trig identity:
`sin x + cos x = sqrt2sin (x + pi/4) = 1`
`sin (x + pi/4) = 1/sqrt2 = sqrt2/2`
The trig unit circle gives 2 solution arcs:
a. `x + pi/4 = pi/4`
`x = 0 + 2kpi`
b. `x + pi/4 = (3pi)/4`
`x = (3pi)/4 - pi/4 = pi/2` (rejected because of condition (1))
Answers for `(0, 2pi)`
x = 0 and `x = 2pi`
Check.
if x = 0 --> `1/cos x = 1` --`sin x/cos x = 0` --> 1 = 0 + 1. OK