Question

Tan x + tan(x +pi/4) = 1.
help me fine x ..​

Answer 1

Recall that

`\tan(x+y)=(\tan x+\tan y)/(1-\tan x\tan y)`

and that
`\tan\frac\pi4=1`. So we have

`\tan x+\tan\left(x+\frac\pi4\right)=1`

`\implies\tan x+(\tan x+\tan\frac\pi4)/(1-\tan x\tan\frac\pi4)=1`

`\implies\tan x+(\tan x+1)/(1-\tan x)=1`

`\implies(\tan x(1-\tan x)+\tan x+1)/(1-\tan x)=1`

`\implies(1+2\tan x-\tan^2x)/(1-\tan x)=1`

As long as
`\tan x\\eq1`, we can write

`\implies1+2\tan x-\tan^2x=1-\tan x`

`\implies3\tan x-\tan^2x=0`

`\implies\tan x(3-\tan x)=0`

`\implies\tan x=0\text{ or }\tan x=3`

`\implies\boxed{x=n\pi\text{ or }x=\tan^(-1)3+n\pi}`

where
`n` is any integer.