Evaluate $\tan 195^{\circ}$ without using the calculator How to evaluate $\tan 195^{\circ}$ without using the calculator, and how to give the answer in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers?

The Tangent Identities:
$$\tan (2\theta) = \frac{2\tan\theta}{1-\tan^2\theta} $$
$$\tan (\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1-\tan\alpha\tan\beta} $$
$$\tan (\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1+\tan\alpha\tan\beta} $$
$$\tan 195^\circ = \tan (180^\circ + 15^\circ)$$
$$\tan(180^\circ) = 0$$
$$\frac{0 + \tan(15^\circ)}{1 - 0 (\tan(15^\circ))} = \tan(15^\circ)$$
$$\tan(15^\circ) = \tan(45^\circ - 30^\circ)$$
$$\tan (45^\circ - 30^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1+\tan(45^\circ)\tan(30^\circ)} $$
$$\tan(45^\circ) = 1$$
$$\tan(30^\circ) = \frac{1}{\sqrt3}$$
$$\frac{1 - \frac{1}{\sqrt3}}{1+\frac{1}{\sqrt3}}$$
