即三角函数中的一组恒等式： [1]
推导过程
对于（1）至（4），可用积化和差公式推导，也可以由和角公式得到，以下用和角公式证明之。
由和角公式有，
两式相加、减便可得到上面的公式（1）、（2），同理可证明公式（3）、（4）。
对于（5）、（6），有
对于（7）、（8）、（9）、（10），也可用类似的方法推出。
证毕。
下面不加推导地给出几个公式。对于正余弦平方的减法，同样有和差化积公式： [1]
和差化积公式的形式比较复杂，记忆中以下几个方面是难点，下面指出了各自的简单记忆方法。
如何只记两个公式甚至一个
我们可以只记上面四个公式的第一个和第三个。
第二个公式中的
，即
，这就可以用第一个公式。
同理，第四个公式中，
，这就可以用第三个公式解决。
如果对诱导公式足够熟悉，可以在运算时把余弦全部转化为正弦，那样就只记住第一个公式就行了。
用的时候想得起一两个就行了。
结果乘以2
这一点最简单的记忆方法是通过三角函数的值域判断。正弦和余弦的值域都是[-1,1]，其积的值域也应该是[-1,1]，而和差的值域却是[-2,2] ，因此乘以2是必须的。
也可以通过其证明来记忆，因为展开两角和差公式后，未抵消的两项相同而造成有系数2，如：
故最后需要乘以2。
只有同名三角函数能和差化积
无论是正弦函数还是余弦函数，都只有同名三角函数的和差能够化为乘积。这一点主要是根据证明记忆，因为如果不是同名三角函数，两角和差公式展开后乘积项的形式都不同，就不会出现相抵消和相同的项，也就无法化简下去了。
（一）
正加正，正在前，
余加余，余并肩。
正减正，余在前，
余减余，负正弦。
（反之亦然）
（二）
帅+帅=帅哥，
帅-帅=哥帅，
哥+哥=哥哥，
哥-哥=负嫂嫂。
（反之亦然）
（三）
口口之和仍口口，
赛赛之和赛口留，
口口之差负赛赛，
赛赛之差口赛收。
（四）
正和正在先，
正差正后迁，
余和一色余，
余差翻了天。
（五）
正弦加正弦，正弦在前面，
正弦减正弦，余弦在前面，
余弦加余弦，余弦全部见，
余弦减余弦，负正弦来见。
声明: 本文采用 BY-NC-SA 协议进行授权 | YueTa
转载请注明转自《三分钟记住和差化积公式》
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