Make $\sin(x)-x\cos(x)$ beautiful?

Question

When computing a Fourier series I came across a term like $$\sin(x)-x\cos(x)$$ Is there a way to reduce this expression, e.g. to only $sin$ or $cos$?

My final series looks like this: $$ f(t) = \dfrac{4U}{T^2\omega^2n^2} \sum^\infty_{n=1} \left( \sin(n\omega T/2) - n\omega T/2 \cdot \cos(n\omega T/2) \right) \sin(n\omega t) $$

Answer 1

You can write

$$\sin(x)-x\cos(x)=\sqrt{1+x^2}\left(\frac1{\sqrt{1+x^2}}\sin(x)-\frac{x}{\sqrt{1+x^2}}\cos(x)\right)$$

So:

$$\forall x\in\mathbb{R},\ \exists\theta_x\in[-\pi,\pi],\ \cos(\theta_x)= \frac{1}{\sqrt{1+x^2}},\ \sin(\theta_x)=-\frac x{\sqrt{1+x^2}} $$

Just useful in terms of esthetic.

Answer 2

This may be acceptable. Let $\, z := x - \tan^{-1}(x). \,$ Then $\, \sin(x) - x \cos(x) = \sin(z) / \cos(x-z) $ which is a quotient of $\,\sin\,$ and $\,\cos.$