Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:

$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$

$$\mathcal{F}^s [f (x)]=\int_0^{\infty} f(t) \sin(xt) dt $$

What can we say about following integral ?

$$\int_0^{\infty} \mathcal{F}^c [f (x)] \mathcal{F}^s [g (x)] dx$$

(with for example $f$ and $g$ continuous functions in $L^2$)

Can we express this integral as a simple integral with original functions ? (like in Parseval's equation)