Which equality belongs to: cos^3 + sin^3?

Answer

Dear Julie,

I suspect you are referring to the identities that convert cubes into trigonometric functions of the triple of the argument:

4 Sin[x]³≡3 Sin[x]-Sin[3x]

4 Cos[x]³≡3·cos[x]+Cos[3x]

Both identities together then give you an opportunity to convert the power increases in your expression into multiples of the argument.

Your question illustrates another problem:
Without further specification, multiple answers are even possible. Depending on the intended application, one answer may already be “better” than the other.
This way you can use the identities

a³+B³ ≡ (A+B)-(A²-A B+B²)

and

cos[x]²+Sin[x]² 1

Convert the sum of cubes also to

cos[x]³+Sin[x]³ ≡ (Cos[x]+Sin[x])·(Cos[x]²-Cos[x]·Sin[x]+Sin[x]²) ≡ (Cos[x]+Sin[x])·(1-Cos[x]·Sin[x])

For certain applications, this conversion of a cubic polynomial in terms of Cos[x] and Sin[x] to an equivalent second degree polynomial are also interesting.

Philippe J. Roussel
imec