Since we're on a right triangle, we'll have that:
[tex]\cos (45)=\frac{5}{x}[/tex]
Solving for x,
[tex]\begin{gathered} \cos (45)=\frac{5}{x} \\ \\ \rightarrow x=\frac{5}{\cos(45)} \\ \\ \Rightarrow x=5\text{ }\sqrt[]{2} \end{gathered}[/tex]
Now, using the pythagorean theroem, we can say that:
[tex]5^2+y^2=x^2[/tex]
Solving for y,
[tex]\begin{gathered} 5^2+y^2=x^2 \\ \rightarrow y^2=x^2-5^2 \\ \rightarrow y=\sqrt[]{x^2-5^2} \\ \rightarrow y=\sqrt[]{(5\text{ }\sqrt[]{2})^2-5^2} \\ \rightarrow y=\sqrt[]{50-25} \\ \rightarrow y=\sqrt[]{25} \\ \\ \Rightarrow y=5 \end{gathered}[/tex]
Therefore, we can conclude that:
[tex]\begin{gathered} x=5\text{ }\sqrt[]{2} \\ y=5 \end{gathered}[/tex]
