Show that $J_0(x) + 2\sum_{n = 1}^\infty J_{2n}(x) = 1$

Question

I want to show this property $J_0(x) + 2\sum_{n = 1}^\infty J_{2n}(x) = 1$.

I tried with the generating function

$$\sum_{n=-\infty}^{n = \infty} J_n(x)t^n = \exp\left(\frac{xt}{2} - \frac{x}{2t}\right).$$

But got nothing.