Trigonometric equation with tan, find x

[Guest]

Solve the trigonometric equation.
Find x
[tex]\tan(x+\frac{\pi}{4})+\tan(x-\frac{\pi}{4})=\tan x[/tex]

[shyamjayakannan]

Use the formula [tex]\tan{(a+b)}=\frac{\tan a+\tan b}{1-\tan a\tan b}[/tex].

So, [tex]\displaystyle\tan{\left(x+\frac{\pi}{4}\right)}+\tan{\left(x-\frac{\pi}{4}\right)}=\tan x\Rightarrow\frac{\displaystyle\tan x+\tan\frac{\pi}{4}}{\displaystyle1-\tan x\tan\frac{\pi}{4}}+\frac{\displaystyle\tan x-\tan\frac{\pi}{4}}{\displaystyle1+\tan x\tan\frac{\pi}{4}}=\tan x\Rightarrow\frac{1+\tan x}{1-\tan x}-\frac{1-\tan x}{1+\tan x}=\tan x[/tex]

[tex]\displaystyle\Rightarrow\frac{(1+\tan x)^2-(1-\tan x)^2}{(1-\tan x)(1+\tan x)}=\tan x\Rightarrow\frac{4\tan x}{1-\tan^2x}=\tan x\Rightarrow\tan x\left(\frac{4}{1-\tan^2x}-1\right)=0\Rightarrow\tan x=0[/tex] or [tex]\frac{4}{1-\tan^2x}-1=0[/tex]

[tex]\frac{4}{1-\tan^2x}-1=0\Rightarrow\tan^2x=-3[/tex], which is not possible for real [tex]x[/tex]. So, [tex]\tan x=0[/tex] and now you can find [tex]x[/tex].