The following identities, known as the **harmonic identities**, are very useful in solving certain types of trig. equation.

Try and learn them. However, it is a good exercise to try and prove them.

They are all very similar in the method of proof and are based on the addition identities.

I would encourage you to look at how I have proved one and then try the others as they are all very similar.

### The Harmonic Identities:

- Asin(x) + Bcos(x) ≡ Rsin(x + α)
- Asin(x) – Bcos(x) ≡ Rsin(x – α)
- Acos(x) + Bsin(x) ≡ Rcos(x – α)
- Acos(x) – Bsin(x) ≡ Rcos(x + α)

where R = √ ( A^{2} + B^{2} ) and α = tan^{-1} B/A

Show:

- Asin(x) + Bcos(x) ≡ Rsin(x + α)

Show:

- Asin(x) – Bcos(x) ≡ Rsin(x – α)

Show:

- Acos(x) + Bsin(x) ≡ Rcos(x – α)

Show:

- Acos(x) – Bsin(x) ≡ Rcos(x + α)