Sum-to-Product Identities

1. Sum of sines

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

2. Difference of sines

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

3. Sum of cosines

$\mathrm{cos}\alpha +\mathrm{cos}\beta =2\mathrm{cos}\frac{\alpha +\beta }{2}\mathrm{cos}\frac{\alpha –\beta }{2}$

4. Difference of cosines

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

5. Sum of tangents

$\mathrm{tan}\alpha +\mathrm{tan}\beta =\frac{\mathrm{sin}\left(\alpha +\beta \right)}{\mathrm{cos}\alpha ·\mathrm{cos}\beta }$

6. Difference of tangents

$\mathrm{tan}\alpha –\mathrm{tan}\beta =\frac{\mathrm{sin}\left(\alpha –\beta \right)}{\mathrm{cos}\alpha ·\mathrm{cos}\beta }$

7. Sum of cotangents

$\mathrm{cot}\alpha +\mathrm{cot}\beta =\frac{\mathrm{sin}\left(\beta +\alpha \right)}{\mathrm{sin}\alpha ·\mathrm{sin}\beta }$

8. Difference of cotangents

$\mathrm{cot}\alpha –\mathrm{cot}\beta =\frac{\mathrm{sin}\left(\beta –\alpha \right)}{\mathrm{sin}\alpha ·\mathrm{sin}\beta }$

9. Sum of cosine and sine

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

10. Difference of cosine and sine

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

11. Sum of tangent and cotangent

$\mathrm{tan}\alpha +\mathrm{cot}\beta =\frac{\mathrm{cos}\left(\alpha –\beta \right)}{\mathrm{cos}\alpha ·\mathrm{sin}\beta }$

12. Difference of tangent and cotangent

$\mathrm{tan}\alpha –\mathrm{cot}\beta =–\frac{\mathrm{cos}\left(\alpha +\beta \right)}{\mathrm{cos}\alpha ·\mathrm{sin}\beta }$

Solution:

$\mathrm{sin}4\theta –\mathrm{sin}2\theta =2\mathrm{cos}\left(\frac{4\theta +2\theta }{2}\right)\mathrm{sin}\left(\frac{4\theta –2\theta }{2}\right)$
$=2\mathrm{cos}\left(\frac{6\theta }{2}\right)\mathrm{sin}\left(\frac{2\theta }{2}\right)$
$=2\mathrm{cos}3\theta \mathrm{sin}\theta$