[tex]\begin{gathered} sin(\theta)=\frac{\sqrt{2}}{5} \\ sin(2\theta)=2sin(\theta)cos(\theta) \\ cos(2\theta)=cos^2(\theta)-sin^2(\theta) \\ cos(\theta)=? \\ sin^2(\theta)+cos^2(\theta)=1 \\ cos^2(\theta)=1-sin^2(\theta) \\ cos^2(\theta)=1-(\frac{\sqrt{2}}{5})^2 \\ cos^2(\theta)=1-\frac{2}{25} \\ cos^2(\theta)=\frac{23}{25} \\ cos(\theta)=\sqrt{\frac{23}{25}} \\ cos(\theta)=\frac{\sqrt{23}}{5} \\ Hence \\ s\imaginaryI n(2\theta)=2s\imaginaryI n(\theta)cos(\theta) \\ s\imaginaryI n(2\theta)=(2)(\frac{\sqrt{2}}{5})(\frac{\sqrt{23}}{5}) \\ s\imaginaryI n(2\theta)=\frac{2\sqrt{46}}{25} \\ \\ cos(2\theta)=cos^2(\theta)-s\imaginaryI n^2(\theta) \\ cos(2\theta)=\frac{23}{25}-\frac{2}{25} \\ cos(2\theta)=\frac{21}{25} \end{gathered}[/tex]
