# Product-to-Sum Identities

Angles (arguments of functions): $$\alpha$$, $$\beta$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$
1. Product of sines
$$\sin \alpha \cdot \sin \beta =$$ $$\large\frac{{\cos \left( {\alpha – \beta } \right) – \cos \left( {\alpha + \beta } \right)}}{2}\normalsize$$
2. Product of cosines
$$\cos \alpha \cdot \cos \beta =$$ $$\large\frac{{\cos \left( {\alpha – \beta } \right) + \cos \left( {\alpha + \beta } \right)}}{2}\normalsize$$
3. Product of sine and cosine
$$\sin \alpha \cdot \cos \beta =$$ $$\large\frac{{\sin \left( {\alpha – \beta } \right) + \sin \left( {\alpha + \beta } \right)}}{2}\normalsize$$
4. Product of tangents
$$\tan\alpha \cdot \tan \beta =$$ $$\large\frac{{\tan\alpha + \tan \beta }}{{\cot\alpha + \cot \beta }}\normalsize$$
5. Product of cotangents
$$\cot\alpha \cdot \cot \beta =$$ $$\large\frac{{\cot\alpha + \cot \beta }}{{\tan\alpha + \tan \beta }}\normalsize$$
6. Product of tangent and cotangent
$$\tan\alpha \cdot \cot \beta =$$ $$\large\frac{{\tan\alpha + \cot \beta }}{{\cot\alpha + \tan \beta }}\normalsize$$