## 2016年5月17日火曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (三角関数の諸公式(2倍角、半角)2)

• 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(Edge/Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

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

2. $\begin{array}{l}{\mathrm{cos}}^{4}\alpha -{\mathrm{sin}}^{4}\alpha \\ =\left({\mathrm{cos}}^{2}\alpha -{\mathrm{sin}}^{2}\alpha \right)\left({\mathrm{cos}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha \right)\\ =\mathrm{cos}2\alpha \end{array}$

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

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