[tex]sin\ \alpha=\frac23\\\\\\sin^2\ \alpha+cos^2\ \alpha=1\\\\(\frac23)^2+cos^2\ \alpha=1\\\\\frac49+cos^2\ \alpha=1\\\\cos^2\ \alpha=1-\frac49\\\\cos^2\ \alpha=\frac59\\\\cos^2\ \alpha=\sqrt{\frac59}}\\\\cos\ \alpha=\frac{\sqrt5}{3}\\\\\\tg\ \alpha=\frac{sin\ \alpha}{cos\ \alpha}\\\\tg\ \alpha=\frac23\ :\ \frac{\sqrt5}{3}\\\\tg\ \alpha=\frac23\ \cdot\ \frac3{\sqrt5}\\\\tg\ \alpha=\frac2{\sqrt5}\ \cdot\ \frac{\sqrt5}{\sqrt5}=\frac{2\sqrt5}{5}[/tex]
[tex]\frac{\sqrt{1+tg^2\ \alpha}}{cos\ \alpha}=\frac{\sqrt{1+(\frac{2\sqrt5}5)^2}}{\frac{\sqrt5}3}=\\\\=\frac{\sqrt{1+\frac{4\cdot5}{25}}}{\frac{\sqrt5}{3}}=\frac{\sqrt{1+\frac{20}{25}}}{\frac{\sqrt5}3}=\\\\=\frac{\sqrt{1\frac45}}{\frac{\sqrt5}3}=\frac{\sqrt{\frac95}}{\frac{\sqrt5}3}=\\\\=\frac{\frac3{\sqrt5}}{\frac{\sqrt5}3}=\frac{3}{\sqrt5}\ :\ \frac{\sqrt5}3=\\\\=\frac3{\sqrt5}\ \cdot\ \frac3{\sqrt5}=\frac95=1\frac45[/tex]