Given: tan x + sec x = √3

squaring on both sides

(tan x + sec x)² = √3²

tan²x+sec²x+2 tan x sec x = 3

Also, sec²x - tan²x = 1

tan²x + 1+ tan²x+ 2tan x sec x = 3

2tan²x+2tan x sec x = 3-1

tan²x+tan x sec x = 2/2

tan²x+tan x sec x = 1

tan x sec x = 1 - tan²x

again, squaring on both sides

tan²x sec²x = 1 + tan⁴x - 2 tan²x

(1+ tan²x) tan²x = 1 + tan⁴x - 2 tan²x

Tan⁴x + tan²x = 1 + tan⁴x - 2 tan²x

3 tan²x = 1

tan x = 1/√3

x = π/6.