Let $n-m>=2$ be two fixed natural numbers. Are there any nontrivial rational solutions of the equation $$x^{n-m}=(1+t^m)/(1+t^n)$$ for $x$ and $t$? As particular cases the rational solutions of the equations $x^2=(1+t^2)/(1+t^4)$ and $x^2=(1+t^3)/(1+t^5)$ will be interesting.

Let $n-m \ge 2$ be two fixed natural numbers. Are there any nontrivial rational solutions of the equation $$x^{n-m}=(1+t^m)/(1+t^n)$$ for $x$ and $t$? As particular cases the rational solutions of the equations $x^2=(1+t^2)/(1+t^4)$ and $x^2=(1+t^3)/(1+t^5)$ will be interesting.