Question

tan x + sec x = √3 find the value of x

Answer 1

Answer:

x = 30°

Step-by-step explanation:

We know that

$1+tan^{2} x=sec^{2}x\\\\so\\\\sec x=\sqrt{1+tan^{2} x}$

Use this identity,and replace sec x by the identity

$tan x+ sec x=\sqrt{3}\\tan x+\sqrt{1+tan^{2}x}=\sqrt{3}\\\\tan x-\sqrt{3}=\sqrt{1+tan^{2} x}$

squaring both sides

$tan^{2} x+3-2\sqrt{3}tan\:x=tan^{2} x+1\\\\-2\sqrt{3}tan\:x=-2\\\\tan\:x=\frac{1}{\sqrt{3}}\\\\tan\:x=tan(30°)\\\\x=30°$

Hope it helps you.

Answer 2

Answer:

x = 30°

Step-by-step explanation:

tan x + sec x = √3

=> Sinx/Cosx  + 1/Cosx = √3

=> Sinx + 1 = √3Cosx

Squaring both sides

=> Sin²x + 1 + 2Sinx = 3Cos²x

=> Sin²x + 1 + 2Sinx = 3 - 3Sin²x

=> 4Sin²x + 2Sinx - 2 = 0

Dividing by 2 both sides

=> 2Sin²x + Sinx - 1 = 0

=> 2Sin²x + 2Sinx - Sinx - 1 =0

=> 2Sinx(Sinx + 1) - 1(Sinx + 1) = 0

=> (2Sinx - 1)(Sinx + 1) = 0

=> Sinx = 1/2  or Sinx = -1

=> x = 30°   or x = -270°

but Tanx & Secx not defined for x = -270°

=> x = 30°