Buktikan identitas trigonometri berikut

Trigonometri.

Nomor 1.
Menggunakan rumus perkalian ke penjumlahan.
[tex]\displaystyle \frac{\cos (\alpha +\beta )+\cos (\alpha -\beta )}{\sin (\alpha +\beta )+\sin (\alpha -\beta )}\\ =\frac{2\cos \alpha \cos \beta }{2\sin \alpha \cos \beta }\\ =\cot \alpha [/tex]
Terbukti.

Nomor 2.
[tex]\displaystyle \boxed {\sin A+\sin B=2\sin \left ( \frac{A+B}{2} \right )\cos \left ( \frac{A-B}{2} \right )}\\ \boxed{\cos A+\cos B=2\cos \left ( \frac{A+B}{2} \right )\cos \left ( \frac{A-B}{2} \right )}[/tex]

[tex]\displaystyle \frac{\sin A+\sin 3A}{\cos A+\cos 3A}\\ =\frac{2\sin \left ( \frac{A+3A}{2} \right )\cos \left ( \frac{A-3A}{2} \right )}{2\cos \left ( \frac{A+3A}{2} \right )\cos \left ( \frac{A-3A}{2} \right )}\\ =\tan 2A[/tex]
Terbukti.