(1 + tan x) / (1 + cot x) = 2 : find a numerical value of one trigonometric function of x?

Answer 1

Solve for x.

(1 + tan x) / (1 + cot x) = 2
1 + tan x = 2(1 + cot x) = 2 + 2cot x
1 + tan x = 2 + 2cot x
tan x = 1 + 2cot x
tan²x = tan x + 2
tan²x - tan x - 2 = 0
(tan x - 2)(tan x + 1) = 0
tan x = 2, -1
The solution -1 is rejected. It was an extraneous solution introduced by squaring.

tan x = 2

x = arctan(2)

Answer 2

The answer is tan x = 2 (or cot x = 1/2).

Divide the original equation by tan x on both sides; that multiplies the denominator by tan x; then :

(1 + tan x) / (tan x + 1) = 1 = 2 / tan x, therefore tan x = 2.

You did not ask for the value of x itself, but for what it's worth, it's x = arctan (2) = 63.4349... + 180 n degrees or 1.1071... + nπ radians, for ' n ' any integer.

CHECK: If tan x = 2, cot x = 1/2, and thus the original expression
= (1 + 2) / (1 + (1/2)) = 3 / (1.5) = 2. It checks out!

Live long and prosper.

Answer 3

(1 + tan x) / (1 + cot x) = 2 multiply both sides by 1+cot x
1+tan x=2+2 cot x subtract 2+2 cot x from each side
tan x -1-2 cot x=0 multiply by tan x
tan^2 x- tan x-2=0
(tan x -2)(tan x +1)=0
tam x -2=0
tan x=2 cot x =0.5
x=arctan 2
x=63.4°

tan x+1=0
tan x=-1; cot x =-1
x=arctan -1=-45°

Answer 4

(1 + tan x) / (1 + cot x), 1+cot x ≠ 0
= tan x(cot x +1)/(1+cot x) , take tan x out
= tan x
= 2

x = arctan 2 = 63.43⁰
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x = -45⁰ is not the solution because 1+cot(-45⁰) = 0

Answer 5

Call tanx= z you get (1+z)/(1+1/z)=2 so z=2
tan x= 2 x =1.1071 +kpi rad

Answer 6

tanx/cotx - secx/cosx = sin²x/cos²x - a million/cos²x = (sin²x-a million)/cos²x = -cos²x/cos²x = -a million= 2/cscx <--> cscx = -2 <--> sinx= -a million/2 <--> x= 7/6·pi+ 2·ok·pi with ok integer or x=11/6·pi+ 2·ok· pi Saludos.