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

1 Answer
Nghi N.
Oct 25, 2016

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

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