# Sum-to-Product Identities

Angles (arguments of functions): $$\alpha$$, $$\beta$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$
1. Sum of sines
$$\sin \alpha + \sin \beta =$$ $$2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize$$
2. Difference of sines
$$\sin \alpha – \sin \beta =$$ $$2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize$$
3. Sum of cosines
$$\cos \alpha + \cos \beta =$$ $$2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize$$
4. Difference of cosines
$$\cos \alpha – \cos \beta =$$ $$-2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize$$
5. Sum of tangents
$$\tan\alpha + \tan \beta =$$ $$\large\frac{{\sin \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize$$
6. Difference of tangents
$$\tan\alpha – \tan \beta =$$ $$\large\frac{{\sin \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize$$
7. Sum of cotangents
$$\cot\alpha + \cot \beta =$$ $$\large\frac{{\sin \left( {\beta + \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize$$
8. Difference of cotangents
$$\cot\alpha – \cot \beta =$$ $$\large\frac{{\sin \left( {\beta – \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize$$
9. Sum of cosine and sine
$$\cos\alpha + \sin \alpha =$$ $$\sqrt 2 \cos \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =$$ $$\sqrt 2 \sin\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)$$
10. Difference of cosine and sine
$$\cos\alpha – \sin \alpha =$$ $$\sqrt 2 \sin \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =$$ $$\sqrt 2 \cos\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)$$
11. Sum of tangent and cotangent
$$\tan\alpha + \cot \beta =$$ $$\large\frac{{\cos \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize$$
12. Difference of tangent and cotangent
$$\tan\alpha – \cot \beta =$$ $$-\large\frac{{\cos \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize$$
13. $$1 + \cos \alpha = 2\,{\cos ^2}\large\frac{\alpha }{2}\normalsize$$
14. $$1 – \cos \alpha = 2\,{\sin ^2}\large\frac{\alpha }{2}\normalsize$$
15. $$1 + \sin \alpha =$$ $$2\,{\cos ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)$$
16. $$1 – \sin \alpha =$$ $$2\,{\sin ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)$$