Prove the following vector identities:

a. \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d})-(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})\).

b. \((\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{b} \times \mathbf{d}) \mathbf{c}-(\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}) \mathbf{d}\).