1.

If tan x + sec x = √3, 0 < x < π, then x is equal toA. \(\cfrac{5\pi}6\)B. \(\cfrac{2\pi}3\)C. \(\cfrac{\pi}6\)D.\(\cfrac{\pi}3\)

Answer»

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 tanx

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

Tan4x + tan2x = 1 + tan4x - 2 tan2x

3 tan2x = 1

tan x = 1/√3

x = π/6.



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