Define Bessel function as: $$J_a(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{\Gamma(a+n+1)n!}}\left({{x}\over{2}}\right)^{a+2n}.$$ Where $a$ not an integer.
Need to show $$J_{a+1}(x)J_{-a}(x)+J_a(x)J_{-(a+1)}(x)=-{{2\sin(a\pi)}\over{\pi x}}.$$
Note that the Wronskian of $$W(J_a(x),J_{-a}(x))=-{{2\sin(a\pi)}\over{\pi x}}.$$
We have \begin{align} W(J_a(x),J_{-a}(x))&=J_a(x)J'_{-a}(x)-J'_a(x)J_{-a}(x)=\\ &=J_a(x)\left(J_{-a-1}(x)+\frac{a}{x}J_{-a}(x)\right)-\left(-J_{a+1}(x)+\frac{a}{x}J_a(x)\right)J_{-a}(x)=\\ &=J_a(x)J_{-a-1}(x)+J_{a+1}(x)J_{-a}(x), \end{align} where we have used the differentiation formulas of Bessel functions (see e.g. 10.6.2 here) to pass from the 1st to the second line. Now using the formula for the Wronskian from your question, we easily obtain the seeked identity.
P.S. The differentiation formulas can be proven either directly using series representations or by verifying that e.g. $\displaystyle J'_a(x)-\frac{a}{x}J_a(x)$ satisfies the same differential equation as $-J_{a+1}(x)$ and comparing their behaviour as $x\rightarrow 0$.