Let f: R → R be a twice differentiable function such that ( sin x cos y)(f(2 x+2 y)-f(2 x-2 y))=( cos x sin y)(f(2 x+2 y)+f(2 x-2 y)), for all x, y ∈…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $\displaystyle (\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all $\displaystyle x, y \in \mathbf{R}$. If $\displaystyle f^{\prime}(0)=\frac{1}{2}$, then the value of $\displaystyle 24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :