2020-M2-03

1. $\begin{array}{cl}
   & \tan 3x \\
   = & \tan (x+2x) \\
   = & \dfrac{\tan x + \tan 2x}{1-\tan x\tan 2x} \\
   = & \dfrac{\tan x + \dfrac{2\tan x}{1-\tan^2 x}}{1-\tan x\times \dfrac{2\tan x}{1-\tan^2 x}} \\
   = & \dfrac{\dfrac{\tan x(1-\tan^2 x)+2\tan x}{1-\tan^2 x}}{\dfrac{1-\tan^2 x-2\tan^2 x}{1-\tan^2 x}} \\
   = & \dfrac{3\tan x-\tan^3 x}{1-3\tan^2 x}
   \end{array}$

2. $\begin{array}{cl}
   & \tan x \tan (60^\circ -x)\tan(60^\circ +x) \\
   = & \tan x \times \dfrac{\tan 60^\circ-\tan x}{1+\tan 60^\circ \tan x} \times \dfrac{\tan 60^\circ+\tan x}{1-\tan 60^\circ \tan x} \\
   = & \tan x \times \dfrac{\tan^2 60^\circ -\tan^2 x}{1-\tan^2 60^\circ\tan^2 x} \\
   = & \tan x \times \dfrac{(\sqrt{3})^2-\tan^2 x}{1-(\sqrt{3})^2\tan^2 x} \\
   = & \dfrac{3\tan x-\tan^3 x}{1-3\tan^2 x} \\
   = & \tan 3x
   \end{array}$

$\begin{array}{rcl}
\tan 5^\circ\tan 55^\circ \tan 65^\circ & = & \tan 15^\circ \\
\dfrac{1}{\tan (90^\circ-5^\circ)}\tan 55^\circ \tan 65^\circ & = & \dfrac{1}{\tan (90^\circ-15^\circ)} \\
\dfrac{1}{\tan 85^\circ}\tan 55^\circ \tan 65^\circ & = & \dfrac{1}{\tan 75^\circ} \\
\tan 55^\circ \tan 65^\circ \tan 75^\circ & = & \tan 85^\circ
\end{array}$