2020-M2-03

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

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

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