Solution: The cosine of \(45^\circ\) is given by \(\cos 45^\circ = \frac{\sqrt{2}}{2}\). For a square with side length \(s = 7\) units, the length of the diagonal \(d\) can be calculated using the Pythagorean theorem: \(d = s\sqrt{2}\). Substituting the value of \(s\), we get \(d = 7\sqrt{2}\). Therefore, the length of the diagonal is \(\boxed{7\sqrt{2}}\).

["The Cosine of 45 Degrees and the Diagonal of a Square: A Practical Solution Using the Pythagorean Theorem", "Understanding fundamental trigonometric values is essential in geometry, especially when analyzing shapes like squares. One key property is the relationship between the side length and diagonal in a square, protected by the Pythagorean theorem. Notably, the cosine of (45^\circ) plays a central role in this relationship. This article explains the connection between (\cos 45^\circ), a square with side (s = 7), and the accurate calculation of its diagonal length.", "### The Cosine of 45 Degrees", "Trigonometric values are crucial for computing distances and angles in right triangles. In a 45-45-90 triangle—common in squares—both non-hypotenuse sides are equal, and the hypotenuse is (s\sqrt{2}). The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. For (45^\circ), where the two legs are equal and adjacent to the angle:", "[\n\cos 45^\circ = \frac{\ ext{adjacent leg}}{\ ext{hypotenuse}} = \frac{s}{s\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\n]", "This value confirms that at (45^\circ), the adjacent side is (\frac{\sqrt{2}}{2}) times the hypotenuse, a truth deeply embedded in geometry.", "### Applying It to a Square with Side Length 7 Units", "Now consider a square where each side measures (s = 7) units. To find the diagonal (d) connecting two opposite corners, we apply the Pythagorean theorem. Since the diagonal splits the square into two congruent right triangles, each with legs of length (s), we compute:", "[\nd = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}\n]", "Substituting (s = 7):", "[\nd = 7\sqrt{2}\n]", "This result aligns with both geometric principles and trigonometric identity—since (\cos 45^\circ = \frac{\sqrt{2}}{2}) confirms the ratio present in this diagonal split.", "### Conclusion", "The diagonal length of a square with side (7) units is elegantly expressed as (\boxed{7\sqrt{2}}) units. The cosine of (45^\circ), (\frac{\sqrt{2}}{2}), serves as a foundational constant that supports and validates this geometric calculation. Understanding this connection enhances problem-solving in geometry and reinforces the harmony between trigonometry and spatial reasoning.", "---", "Keywords: cosine of 45 degrees, cos 45°, diagonal of a square, Pythagorean theorem, 45-45-90 triangle, (7\sqrt{2}), geometry solution, trigonometry in squares, side length calculation, square diagonal formula"]