Sharp second-order sinc–tanc Huygens inequalities and matrix extensions

We resolve the second-order remainder problem for the sinc–tanc Huygens-type inequalities, advancing beyond recent optimal first-order estimates. For the classical combinations $$ H(x)={{\,\textrm{sinc}\,}}(2x)+2{{\,\textrm{tanc}\,}}x,\quad G(x)=\frac{1}{{{\,\textrm{sinc}\,}}(2x)}+\frac{2}{{{\,\textrm{tanc}\,}}x},\quad 0<x<\frac{\pi }{2}, $$ we prove that the normalized second-order remainders $$ \frac{\frac{2}{5}x^3\tan x-[H(x)-3]}{x^5\tan x} \quad \text {and}\quad \frac{[G(x)-3]-\frac{4}{15}x^3\tan x}{x^5\tan x} $$ are respectively strictly decreasing and strictly increasing. Overcoming the inherent limitations of local Taylor expansions, we establish explicit alternating-series coefficient identities and uniform ratio bounds to determine the derivative signs globally throughout the interval. This strict monotonicity directly yields sharp two-sided inequalities with the best possible endpoint constants. As analytical consequences, we obtain the complete stratified sign classifications, establish analogous global hyperbolic inequalities, and record direct spectral lifts to Hermitian matrices.