\(\displaystyle{\cos{{2}}}{t}={2}{{\cos}^{{2}}{t}}-{1}\) hence

\(\displaystyle\int{\frac{{{\tan{{t}}}{\cos{{2}}}{t}}}{{{2}}}}\)dt=\int\sin t\cos tdt-\frac{1}{2}\tan tdtask

\(\displaystyle={\frac{{{1}}}{{{2}}}}{{\sin}^{{2}}{t}}-{\frac{{{1}}}{{{2}}}}{\ln{{\left({\sec{{t}}}\right)}}}+{C}\)

\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({{\sin}^{{2}}{t}}+{\ln{{\left({\cos{{t}}}\right)}}}\right)}+{C}\)

\(\displaystyle\int{\frac{{{\tan{{t}}}{\cos{{2}}}{t}}}{{{2}}}}\)dt=\int\sin t\cos tdt-\frac{1}{2}\tan tdtask

\(\displaystyle={\frac{{{1}}}{{{2}}}}{{\sin}^{{2}}{t}}-{\frac{{{1}}}{{{2}}}}{\ln{{\left({\sec{{t}}}\right)}}}+{C}\)

\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({{\sin}^{{2}}{t}}+{\ln{{\left({\cos{{t}}}\right)}}}\right)}+{C}\)