Sum-to-Product Identities

1. Sum of sines

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

2. Difference of sines

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

3. Sum of cosines

$$ \cos \alpha + \cos \beta = 2 \cos \frac {\alpha + \beta}{2} \cos \frac {\alpha - \beta}{2} $$

4. Difference of cosines

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

5. Sum of tangents

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

6. Difference of tangents

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

7. Sum of cotangents

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

8. Difference of cotangents

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

9. Sum of cosine and sine

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

10. Difference of cosine and sine

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

11. Sum of tangent and cotangent

$$ \tan \alpha + \cot \beta = \frac {\cos (\alpha - \beta)}{\cos \alpha \cdot \sin \beta} $$

12. Difference of tangent and cotangent

$$ \tan \alpha - \cot \beta = - \frac {\cos (\alpha + \beta)}{\cos \alpha \cdot \sin \beta} $$

$$ 13.\ 1+ \cos \alpha = 2 \cos^2 \frac {\alpha }{2} $$

$$ 14.\ 1- \cos \alpha = 2 \sin^2 \frac {\alpha }{2} $$

$$ 15.\ 1+ \sin \alpha = 2 \cos^2 \left( \frac {\pi }{4} - \frac {\alpha }{2} \right) $$

$$ 16.\ 1- \sin \alpha = 2 \sin^2 \left( \frac {\pi }{4} - \frac {\alpha }{2} \right) $$

Example:

Write the following difference of sines expression as a product: sin(4θ) − sin(2θ)

Solution:

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

$$ = 2 \cos \left( \frac {6\theta}{2} \right) \sin \left( \frac {2\theta}{2} \right) $$

$$ = 2 \cos 3\theta \sin \theta $$