An Identity Concerning the Riemann Zeta Function

Let $\zeta$ be the Riemann- Zeta function. For any integer, $n \geq 2$, how to prove $$\zeta(2) \zeta(2n-2) + \zeta(4)\zeta(2n-4) + \cdots + \zeta(2n-2)\zeta(2) = \Bigl(n + \frac{1}{2}\Bigr)\zeta(2n)$$

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