Mathematics Question from JEE Main 2025 · 2 April, Shift 1

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that

$$(\sin x \cos y)\big(f(2x + 2y) - f(2x - 2y)\big) = (\cos x \sin y)\big(f(2x + 2y) + f(2x - 2y)\big)$$

for all $x, y \in \mathbb{R}$. If $f'(0) = \frac{1}{2}$, then the value of $24 f''\left(\frac{5\pi}{3}\right)$ is: