Find all solutions in the interval [0, 2π). tan x + sec x = 1

Use the identity

sec^2x = 1 + tan^2 x

- so sec x = sqrt (1 + tan^2 x) then:-

tan x + sqrt (1 + tan^2 x) = 1

sqrt (1 + tan^2 x) = 1 - tan x

1 + tan^2 x = 1 + tan^2x - 2 tan x

0 = - 2 tanx

tan x = 0

x = 0, π

π is an extraneous root because sec 180 = - 1

So the answer is 0 degrees