Some help on trigonometric function in sympy please

[Javier]

Hi all,

Anyone knows if there is a simple way of doing an expansion of
sin(x)**p and cos(x)**p as a sum of cos(ix) or sin(ix) in sympy. I
currently can do it in maple with the combine command but I want to
use sympy.


The idea is that for example:

sin(x)**5 = 1/16 sin(5 x) - 5/16 sin(3 x) + 5/8 sin(x)
cos(x)**5 = 1/16 cos(5 x) + 5/16 cos(3 x) + 5/8 cos(x)

Can anyone help me please?

Thanks in advance

[Aaron S. Meurer]

Unfortunately, this simplification is not yet implemented in SymPy (though the reverse is, see expand(trig=True)). See http://code.google.com/p/sympy/issues/detail?id=1590.

It shouldn't be too hard to implement, though, you just need to implement the series expansions given at http://en.wikipedia.org/wiki/Trig_identities#Power-reduction_formula.

Actually, there is another way that you can do it, which involves taking advantage of the complex exponential form of sine and cosine. You would do something like this:

In [3]: (cos(x)**5).rewrite(exp).expand().rewrite(cos)
Out[3]:
cos(5⋅x) 5⋅cos(x) 5⋅cos(3⋅x)
──────── + ──────── + ──────────
16 8 16

Aaron Meurer

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