# Can you express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos?

I was going through some of my notes and found they said we can express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos. However, I can't see how this is possible, and can only express it as:

$\cos(\frac{x}{2})\sin(x) = \sin(\frac{3x}{2})+\sin(\frac{x}{2})$

So is there another way to express the original expression, so that we would be summing up terms like $\sin(x), \sin(2x), \sin(3x)...$ and $\cos(\frac{x}{2}), \cos(\frac{3x}{2}), \cos(\frac{5x}{2})...$?

• Perhaps this relates to the Dirichlet kernel? – awwalker Mar 22 '13 at 7:09
• That should be $2\cos(\frac{x}{2})\sin(x) = \sin(\frac{3x}{2})+\sin(\frac{x}{2})$. – Inceptio Mar 22 '13 at 7:34

The left hand side is an odd function. Hence, no linear combination involving $\cos a x$ may be equal, whatever the value of $a \neq 0$.
$$\sin C+\sin D= 2 \sin \left(\frac{(C+D)}{2}\right)\cdot\cos\left(\frac{C-D}{2}\right)$$
• @Ryker, And there is an error in your calculation. It should $2\cos(\frac{x}{2})\sin(x) = \sin(\frac{3x}{2})+\sin(\frac{x}{2})$ – Inceptio Mar 22 '13 at 15:45