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For example, for any prime $p>3$ with $p\\equiv3\\pmod4$ and $a,b\\in\\mathbb Z$ with $p\\nmid ab$, we prove that $$\\det\\left[ 1+\\tan\\pi\\frac{aj^2+bk^2}p \\right]_{1\\le j,k \\le \\frac{p-1}2} = \\begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=-1, \\end{cases}$$ where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. 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For example, for any prime $p>3$ with $p\\equiv3\\pmod4$ and $a,b\\in\\mathbb Z$ with $p\\nmid ab$, we prove that $$\\det\\left[ 1+\\tan\\pi\\frac{aj^2+bk^2}p \\right]_{1\\le j,k \\le \\frac{p-1}2} = \\begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=-1, \\end{cases}$$ where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. 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