The polarization identity holds for any scalar product $\langle \cdot,\cdot \rangle$:
$$\langle x,y \rangle = \frac{1}{4} \big( \langle x+y,x+y \rangle - \langle x-y,x-y \rangle \big).$$
In $\mathbb{R}$ this equality boils down to
$$x \cdot y = \frac{1}{4} \big( (x+y)^2-(x-y)^2 \big). \tag{1}$$
One important application in stochastic calculus is a generalization of Itô's isometry: In fact, using $(1)$, it follows easily that
$$\mathbb{E} \left( \left[ \int_0^t f(s) \, dB_s \right]^2 \right) = \mathbb{E} \int_0^t f(s)^2 \, ds$$
implies
$$\mathbb{E} \left( \int_0^t f(s) \, dB_s \cdot \int_0^t g(s) \, dB_s \right) = \mathbb{E} \int_0^t f(s) \cdot g(s) \, ds.$$