Sum to product formulas

Write the products of trigonometric functions as sum of trigonometry expressions.

Sum of sines

$\sin \alpha + \sin \beta = 2 \cdot \sin\dfrac{\alpha+\beta}{2}\cdot \cos \dfrac{\alpha-\beta}{2}$

Difference of sines

$\sin \alpha - \sin \beta = 2\cdot \sin\dfrac{\alpha - \beta}{2} \cdot \cos \dfrac{\alpha+\beta}{2}$

Sum of cosines

$\cos\alpha + \cos\beta = 2 \cdot \cos \dfrac{\alpha+\beta}{2} \cdot \cos \dfrac{\alpha-\beta}{2}$

Difference of cosines

$\cos\alpha - \cos\beta = -2 \cdot \sin \dfrac{\alpha+\beta}{2} \cdot \sin \dfrac{\alpha-\beta}{2}$

Sum of tangents

$\tan \alpha + \tan \beta = \dfrac{\sin(\alpha+\beta)}{\cos\alpha \cdot \cos \beta}$

Difference of tangents

$\tan \alpha - \tan \beta = \dfrac{\sin(\alpha-\beta)}{\cos\alpha \cdot \cos \beta}$

Sum of cotangents

$\cot \alpha + \cot \beta = \dfrac{\sin(\beta + \alpha)}{\sin\alpha \cdot \sin\beta}$

Difference of cotangents

$\cot \alpha - \cot \beta = \dfrac{\sin(\beta - \alpha)}{\sin\alpha \cdot \sin\beta}$

$\cos\alpha + \sin\alpha = \sqrt{2} \cos\left( \frac{\pi}{4}-\alpha\right)$

10.

$\cos\alpha - \sin\alpha = \sqrt{2} \sin\left( \frac{\pi}{4}-\alpha\right)$