Use Euler’s Formula to Prove Trigonometric Identities

Lei Yan

2019/02/13

今天做PRML习题的时候看到的,用欧拉公式\[ e^{ix} = \text{cos}(x) + i\text{sin}(x) \] 来证明下面三个等式: \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \tag{1} \] \[ \text{cos}(x - y) = \text{cos}(x)\text{cos}(y) + \text{sin}(x)\text{sin}(y) \tag{2} \] \[ \text{sin}(x - y) = \text{sin}(x)\text{cos}(y) - \text{cos}(x)\text{sin}(y) \tag{3} \] 证明非常简单,利用 \[ e^{ix}e^{-ix} = 1 \] 可以证明等式\((1)\)。 利用 \[ \text{cos}(x - y) = \mathbf{Re}(e^{i(x-y)}) \quad \text{sin}(x - y) = \mathbf{Im}(e^{i(x-y)}) \] 可以证明等式\((2),(3)\)