## 2016年4月29日金曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (正弦・余弦の加法定理3)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

1. $\begin{array}{l}\left(\mathrm{sin}\alpha \mathrm{cos}\beta +\mathrm{cos}\alpha \mathrm{sin}\beta \right)\left(\mathrm{sin}\alpha \mathrm{cos}\beta -\mathrm{cos}\alpha \mathrm{sin}\beta \right)\\ ={\mathrm{sin}}^{2}\alpha {\mathrm{cos}}^{2}\beta -{\mathrm{cos}}^{2}\alpha {\mathrm{sin}}^{2}\beta \\ ={\mathrm{sin}}^{2}\alpha \left(1-{\mathrm{sin}}^{2}\beta \right)-\left(1-{\mathrm{sin}}^{2}\alpha \right){\mathrm{sin}}^{2}\beta \\ ={\mathrm{sin}}^{2}\alpha -{\mathrm{sin}}^{2}\beta \\ =\left(1-{\mathrm{cos}}^{2}\alpha \right)-\left(1-{\mathrm{cos}}^{2}\beta \right)\\ ={\mathrm{cos}}^{2}\beta -{\mathrm{cos}}^{2}\alpha \end{array}$

2. $\begin{array}{l}\left(\mathrm{cos}\alpha \mathrm{cos}\beta -\mathrm{sin}\alpha \mathrm{sin}\beta \right)\left(\mathrm{cos}\alpha \mathrm{cos}\beta +\mathrm{sin}\alpha \mathrm{sin}\beta \right)\\ ={\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}\beta -{\mathrm{sin}}^{2}\alpha {\mathrm{sin}}^{2}\beta \\ ={\mathrm{cos}}^{2}\alpha \left(1-{\mathrm{sin}}^{2}\beta \right)-\left(1-{\mathrm{cos}}^{2}\alpha \right){\mathrm{sin}}^{2}\beta \\ ={\mathrm{cos}}^{2}\alpha -{\mathrm{sin}}^{2}\beta \\ =\left(1-{\mathrm{sin}}^{2}\alpha \right)-\left(1-{\mathrm{cos}}^{2}\beta \right)\\ ={\mathrm{cos}}^{2}\beta -{\mathrm{sin}}^{2}\alpha \end{array}$