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Sum-to-Product Identities

Sum-to-Product Identities

1. Sum of sines

sinα+sinβ=2sinα+β2cosα–β2 \sin \alpha + \sin \beta = 2 \sin \frac {\alpha + \beta}{2} \cos \frac {\alpha – \beta}{2}

2. Difference of sines

sinα–sinβ=2cosα+β2sinα–β2 \sin \alpha – \sin \beta = 2 \cos \frac {\alpha + \beta}{2} \sin \frac {\alpha – \beta}{2}

3. Sum of cosines

cosα+cosβ=2cosα+β2cosα–β2 \cos \alpha + \cos \beta = 2 \cos \frac {\alpha + \beta}{2} \cos \frac {\alpha – \beta}{2}

4. Difference of cosines

cosα–cosβ=–2sinα+β2sinα–β2 \cos \alpha – \cos \beta = -2 \sin \frac {\alpha + \beta}{2} \sin \frac {\alpha – \beta}{2}

5. Sum of tangents

tanα+tanβ=sin(α+β)cosα·cosβ \tan \alpha + \tan \beta = \frac {\sin (\alpha + \beta)}{\cos \alpha \cdot \cos \beta}

6. Difference of tangents

tanα–tanβ=sin(α–β)cosα·cosβ \tan \alpha – \tan \beta = \frac {\sin (\alpha – \beta)}{\cos \alpha \cdot \cos \beta}

7. Sum of cotangents

cotα+cotβ=sin(β+α)sinα·sinβ \cot \alpha + \cot \beta = \frac {\sin (\beta + \alpha )}{\sin \alpha \cdot \sin \beta}

8. Difference of cotangents

cotα–cotβ=sin(β–α)sinα·sinβ \cot \alpha – \cot \beta = \frac {\sin (\beta – \alpha )}{\sin \alpha \cdot \sin \beta}

9. Sum of cosine and sine

cosα+sinα=2cosπ4–α=2sinπ4+α \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

cosα–sinα=2sinπ4–α=2cosπ4+α \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

tanα+cotβ=cos(α–β)cosα·sinβ \tan \alpha + \cot \beta = \frac {\cos (\alpha – \beta)}{\cos \alpha \cdot \sin \beta}

12. Difference of tangent and cotangent

tanα–cotβ=–cos(α+β)cosα·sinβ \tan \alpha – \cot \beta = – \frac {\cos (\alpha + \beta)}{\cos \alpha \cdot \sin \beta}
13. 1+cosα=2cos2α2 13.\ 1+ \cos \alpha = 2 \cos^2 \frac {\alpha }{2}
14. 1–cosα=2sin2α2 14.\ 1- \cos \alpha = 2 \sin^2 \frac {\alpha }{2}
15. 1+sinα=2cos2π4–α2 15.\ 1+ \sin \alpha = 2 \cos^2 \left( \frac {\pi }{4} – \frac {\alpha }{2} \right)
16. 1–sinα=2sin2π4–α2 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:

sin4θ–sin2θ=2cos4θ+2θ2sin4θ–2θ2 \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)
=2cos6θ2sin2θ2 = 2 \cos \left( \frac {6\theta}{2} \right) \sin \left( \frac {2\theta}{2} \right)
=2cos3θsinθ = 2 \cos 3\theta \sin \theta

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