My (very simple) question to a friend was how do I prove the following using basic trig principles:
$\tan75^\circ = 2 + \sqrt{3}$
He gave this proof (via a text message!)
$1. \tan75^\circ$
$2. = \tan(60^\circ + (30/2)^\circ)$
$3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 – \tan60^\circ \tan(30/2)^\circ) $
$4. \tan (30/2)^\circ = \dfrac{(1 – \cos30^\circ)}{ \sin30^\circ}$
Can this be explained more succinctly as I'm new to trigonometry and a little lost after (2.) ?
EDIT
Using the answers given I'm almost there:
- $\tan75^\circ$
- $\tan(45^\circ + 30^\circ)$
- $\sin(45^\circ + 30^\circ) / \cos(45^\circ + 30^\circ)$
- $(\sin30^\circ.\cos45^\circ + \sin45^\circ.\cos30^\circ) / (\cos30^\circ.\cos45^\circ – \sin45^\circ.\sin30^\circ)$
- $\dfrac{(1/2\sqrt{2}) + (3/2\sqrt{2})}{(3/2\sqrt{2}) – (1/2\sqrt{2})}$
- $\dfrac{(1 + \sqrt{3})}{(\sqrt{3}) – 1}$
- multiply throughout by $(\sqrt{3}) + 1)$
Another alternative approach:
- $\tan75^\circ$
- $\tan(45^\circ + 30^\circ)$
- $\dfrac{\tan45^\circ + \tan30^\circ}{1-\tan45^\circ.\tan30^\circ}$
- $\dfrac{1 + 1/\sqrt{3}}{1-1/\sqrt{3}}$
- at point 6 in above alternative
Best Answer
A proof without words (but it uses some geometry). Is that OK?