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This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$

The OP has changed the problem, thus invalidating the previous answer. After the change, the answer becomes positive. Indeed, the problem is a real algebraic one, since, for any real $u$ and $v$, one can find a real $t$ such that $u=\cos2t$ and $v=\sin2t$ iff $u^2+v^2=1$. Any real algebraic problem can be solved purely algorithmically. In Mathematica, such algorithms are realized via commands Reduce, FindInstance, etc.

In particular, for the function $g$ defined by the Mathematica code in the OP, Mathematica confirms (in about 2.8 sec) that the set of all real triples $(x,y,t)$ such that $g(x,y,t)<1$ is empty:

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$

The OP has changed the problem, thus invalidating the previous answer. After the change, the answer becomes positive. Indeed, the problem is a real algebraic one, since, for any real $u$ and $v$, one can find a real $t$ such that $u=\cos2t$ and $v=\sin2t$ iff $u^2+v^2=1$. Any real algebraic problem can be solved purely algorithmically. In Mathematica, such algorithms are realized via commands Reduce, FindInstance, etc.

In particular, for the function $g$ defined by the Mathematica code in the OP, Mathematica confirms (in about 2.8 sec) that the set of all real triples $(x,y,t)$ such that $g(x,y,t)<1$ is empty:

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$

The OP has changed the problem, thus invalidating the previous answer. After the change, the answer becomes positive. Indeed, the problem is a real algebraic one, since, for any real $u$ and $v$, one can find a real $t$ such that $u=\cos2t$ and $v=\sin2t$ iff $u^2+v^2=1$. Any real algebraic problem can be solved purely algorithmically. In Mathematica, such algorithms are realized via commands Reduce, FindInstance, etc.

In particular, for the function $g$ defined by the Mathematica code in the OP, Mathematica confirms (in about 2.8 sec) that the set of all real triples $(x,y,t)$ such that $g(x,y,t)<1$ is empty:

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots<1.$$

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$