###### tags: `數學` `數學教室` `國高中數學` `淺談` `三角函數`
# 淺談 - 積化和差、和差化積公式
## 目標
- $sin(x+y) + sin(x-y) = 2\ sin(x)\ cos(y)$
- $cos(x+y) + cos(x-y) = 2\ cos(x)\ cos(y)$
- $cos(x+y) - cos(x-y) = -2\ sin(x)\ sin(y)$
## 證明
我們用代數的方法，試著展開左式。

和差角公式：
- $sin(x+y) = sin(x)\ cos(y) + cos(x)\ sin(y)$
- $sin(x-y) = sin(x)\ cos(y) - cos(x)\ sin(y)$
- $cos(x+y) = cos(x)\ cos(y) - sin(x)\ sin(y)$
- $cos(x-y) = cos(x)\ cos(y) + sin(x)\ sin(y)$

我們將前兩式跟後兩式各自相加，得到：

\begin{align*}
sin(x+y) + sin(x-y) &= [sin(x)\ cos(y) + cos(x)\ sin(y)] + [sin(x)\ cos(y) - cos(x)\ sin(y)] \\
&= 2\ sin(x)\ cos(y)
\end{align*}

\begin{align*}
cos(x+y) + cos(x-y) &= [cos(x)\ cos(y) - sin(x)\ sin(y)] + [cos(x)\ cos(y) + sin(x)\ sin(y)] \\
&= 2\ cos(x)\ cos(y)
\end{align*}

相減則可以得到 Goal 的第三式

{%hackmd XLFYqwvTTaiUGmzsCiFAIw %}