If tan x + sec x = √ 3, 0x < π, then x is equal to A. 5π/6 B. 2π/3 C. π/6 D.π/3

Question

If tan x + sec x = √3, 0 < x < π, then x is equal to

A. \(\cfrac{5\pi}6\)

B. \(\cfrac{2\pi}3\)

C. \(\cfrac{\pi}6\)

D.\(\cfrac{\pi}3\)

Answer 1

Correct option is C. \(\cfrac{\pi}6\) 

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 tan² x

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

Tan⁴x + tan²x = 1 + tan⁴x - 2 tan²x

3 tan²x = 1

tan x = 1/√3

x = π/6.