The cosine of a $45^\circ$ angle is given by the expression $\frac{\sqrt{2}}{2}$, which is approximately $0.7071$.

To grasp why this is true, let’s explore some fundamental concepts in trigonometry. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to that of the hypotenuse. In the case of a $45^\circ$ angle, we can consider an isosceles right-angled triangle, where the two legs (the sides that are not the hypotenuse) are of equal length.

Visualize a right-angled triangle where the two non-right angles are both $45^\circ$. If we assign a length of $1$ unit to each of the equal sides, we can apply the Pythagorean theorem to determine the length of the hypotenuse. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). This gives us the following equation:

Substituting the lengths of the sides:

Simplifying further, we find:

Taking the square root of both sides yields:

Now, the cosine of a $45^\circ$ angle can be calculated as the length of the adjacent side (which is $1$) divided by the length of the hypotenuse (which is $\sqrt{2}$):

To simplify this expression, we can multiply both the numerator and denominator by $\sqrt{2}$:

Thus, the exact value of the cosine of a $45^\circ$ angle is $\frac{\sqrt{2}}{2}$. In its decimal form, this value approximates to $0.7071$. This value is particularly significant in various applications of trigonometry, including solving problems related to right-angled triangles and analyzing wave functions in physics.