Product-to-Sum Identities

$$ 1. \ \sin \alpha \cdot \sin \beta = \frac 12 \left[\cos (\alpha – \beta) – \cos (\alpha + \beta)\right] $$

$$ 2. \ \cos \alpha \cdot \cos \beta = \frac 12 \left[\cos (\alpha – \beta) + \cos (\alpha + \beta)\right] $$

$$ 3. \ \sin \alpha \cdot \cos \beta = \frac 12 \left[\sin (\alpha + \beta) + \sin (\alpha – \beta)\right] $$

$$ 4. \ \cos \alpha \cdot \sin \beta = \frac 12 \left[\sin (\alpha + \beta) – \sin (\alpha – \beta)\right] $$

$$ 5. \ \tan \alpha \cdot \tan \beta= \frac {(\tan \alpha + \tan \beta)}{(\cot \alpha + \cot \beta)} $$

$$ 6. \ \cot \alpha \cdot \cot \beta= \frac {(\cot \alpha + \cot \beta)}{(\tan \alpha + \tan \beta)} $$

$$ 7. \ \tan \alpha \cdot \cot \beta= \frac {(\tan \alpha + \cot \beta)}{(\cot \alpha + \tan \beta)} $$

Example:

Write cos 3x cos 2x as a sum

Solution:

$$ \cos \alpha \cdot \cos \beta = \frac 12 [\cos (\alpha – \beta) + \cos (\alpha + \beta)] $$

$$ \cos 3x \cdot \cos 2x = \frac 12 [\cos (3x – 2x) + \cos (3x + 2x)] $$

$$ \cos 3x \cdot \cos 2x = \frac 12 [\cos x + \cos 5x] $$

$$ \cos 3x \cdot \cos 2x = \frac {\cos x}{2} + \frac {\cos 5x}{2} $$