Prove that $\tan(75^\circ) = 2 + \sqrt{3}$
A proof without words (but it uses some geometry). Is that OK?
The formula you want to see is: $\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}$ for any degrees $x$ and $y$.
On the other hand, proving this tangent equality from the formulas $\sin(x+y)=\sin(x)\cos(y)+\sin(y)\cos(x)$ and $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$ will be a good exercise for a beginner.
You can rather use $\tan (75)=\tan(45+30)$ and plug into the formula by Metin. Cause: Your $15^\circ$ is not so trivial.