Maximum Value

$\begin{aligned} y & = \sin x \left(\sin^3 x + 3\right) + \cos x \left(\cos^3 x + 4\right) + \frac{\sin^2 2x}2 + 4\cos x + 5 \\ & = \sin^4 x + 3\sin x + \cos^4 x + 4\cos x + 2\sin^2 x \cos^2 x + 4\cos x + 5 \\ & = {\color{#3D99F6} \sin^4 x + 2\sin^2 x \cos^2 x + \cos^4 x} + 3\sin x + 8\cos x + 5 \\ & = {\color{#3D99F6}\left( \sin^2 x + \cos^2 x \right)^2} + 3\sin x + 8\cos x + 5 \\ & = {\color{#3D99F6}1} + 3\sin x + 8\cos x + 5 \\ & = 3\sin x + 8\cos x + 6 \\ & = \sqrt{73}\left(\frac 3{\sqrt{73}} \sin x + \frac 8{\sqrt{73}}\cos x \right) + 6 \\ & = \sqrt{73}\sin \left(x+ \tan^{-1} \frac 83\right) + 6 \end{aligned}$

Since $\max \left(\sin \left(x+ \tan^{-1} \frac 83\right) \right) = 1$ , $\implies \max (y) = \sqrt{73}+6 \approx \boxed{14.544}$ .