Given that $\tan 2x+\tan x=0$, show that $\tan x=0$ Given that $\tan 2x+\tan x=0$, show that $\tan x=0$
Using the Trigonometric Addition Formulae,
\begin{align}
\tan 2x & = \frac{2\tan x}{1-\tan ^2 x} \\
\Rightarrow \frac{2\tan x}{1-\tan ^2 x}+\tan x & = 0 \\ 
\ 2\tan x+\tan x(1-\tan ^2 x) & = 0 \\
2+1-\tan ^2 x & = 0 \\ 
\tan ^2 x & = 3
\end{align}
This is as far as I can get, and when I look at the Mark Scheme no other Trignometric Identities have been used. Thanks

Notice that the following conditions are equivalent:
\begin{align*}
\tan x + \tan y &= 0\\
\tan x &= -\tan y\\
\tan x &= \tan(-y)\\
x &= -y + k\pi\\
x+y&= k\pi
\end{align*}
If we use $y=2x$ we get
\begin{align*}
3x &= k\pi\\
 x &= k\frac\pi3
\end{align*}
