The Student t ratio with \(n\) degrees of freedom is denoted \(T_n\), where \(T_n=\frac{Z}{\sqrt{\frac{U}{n}}}\), \(Z\) is a standard normal random variable and \(U\) is a \(\chi^2\) random variable independent of \(Z\) with \(n\) degrees of freedom.
Because \(T_n^2= \frac{Z^2}{U/n}\) has an \(F\) distribution with \(1\) and \(n\) df, then, \[f_{T_n^2}(t)=\frac{\Gamma(\frac{1+n}{2})}{\Gamma(\frac{1}{2})\Gamma(\frac{n}{2})}\frac{n^{\frac{n}{2}}t^{-\frac{1}{2}}}{(n+t)^{\frac{1+n}{2}}},\quad t>0\]
Then, \[\begin{align} f_{T_n}(t)&=\frac{d}{dt}F_{T_n}(t)\\ &=\frac{d}{dt}P(T_n\le t)\\ &=\frac{d}{dt}(\frac{1}{2}+P(0\le T_n\le t))\\ &=\frac{d}{dt}(\frac{1}{2}+\frac{1}{2}P(-t\le T_n\le t))\quad (t>0)\\ &=\frac{d}{dt}(\frac{1}{2}+\frac{1}{2}P(T_n^2\le t^2))\\ &=\frac{d}{dt}(\frac{1}{2}+\frac{1}{2}F_{T_n^2}(t^2))\\ &=t\cdot f_{T_n^2}(t^2)\\ &=t\cdot \frac{\Gamma(\frac{1+n}{2})}{\Gamma(\frac{1}{2})\Gamma(\frac{n}{2})}\frac{n^{\frac{n}{2}}t^{-1}}{(n+t^2)^{\frac{1+n}{2}}}\\ &=\frac{\Gamma(\frac{1+n}{2})}{\Gamma(\frac{1}{2})\Gamma(\frac{n}{2})}\frac{1}{\sqrt{n}}\frac{1}{(1+\frac{t^2}{n})^{\frac{1+n}{2}}}\\ &=\frac{\Gamma(\frac{1+n}{2})}{\Gamma(\frac{n}{2})}\frac{1}{\sqrt{n\pi}}\frac{1}{(1+\frac{t^2}{n})^{\frac{1+n}{2}}} \end{align}\]