Variance

$$\sigma^2 \equiv \cfrac{\sum_{i=1}^n(x_i-\mu)^2}{n}$$

$$\sigma^2 = \cfrac{\sum_{i=1}^n(x_i^2-2\mu x_i+\mu^2)}{n}$$

$$\sigma^2 = \cfrac{\sum_{i=1}^nx_i^2}{n} - \cfrac{2\mu\sum_{i=1}^n x_i}{n}+\cfrac{\sum_{i=1}^n \mu^2}{n}$$

$$\sigma^2 = \cfrac{\sum_{i=1}^nx_i^2}{n} - 2\mu^2 + \mu^2 = \frac1n\sum_{i=1}^nx_i^2 - \mu^2 \\\square$$

If you are familiar with Expectation, this is equivalent to

$$Var(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2$$

For more details on the proof above, here is a Khan Academy video on the topic.