Trail to infinity

Algebra Level 2

Let \(S\) denote the infinite sum

\[ \dfrac{1}{1^2} + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + \cdots .\]

Find the value of

\[ \dfrac{1}{1^2} + \dfrac{1}{3^2} + \dfrac{1}{5^2} + \dfrac{1}{7^2} + \cdots \]

in terms of \(S\).

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