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Understanding special angles in trigonometry is crucial for simplifying trigonometric expressions. Special angles, also referred to as 'standard angles', are angles that have exact, well-known trigonometric values. These include angles like 0°, 30°, 45°, 60°, and 90°.

For instance, a 45° angle is significant because in an isosceles right triangle (where the two legs are of equal length), the trigonometric ratios for sine and cosine are equal. The sine and cosine of a 45° angle are both \(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\), as shown in the exercise. This simplifies the process of finding trigonometric values for such angles without the need for a calculator or a trigonometric table. It's important to memorize these values for quick recall during calculations.

The unit circle is a fundamental tool in understanding trigonometry and helps explain why the sine and cosine of 45° are \(\frac{\sqrt{2}}{2}\). A unit circle is a circle with a radius of one unit, centered at the origin of a coordinate system. Each point on the unit circle represents the endpoint of a radius forming an angle with the positive x-axis.

The x-coordinate of the point where the radius intersects the circle represents the cosine of that angle, while the y-coordinate represents the sine. For any angle θ, the coordinates can be denoted as (cos θ, sin θ). Hence, on the unit circle, the point at 45° has coordinates \(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\), reflecting the sine and cosine values for that angle.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are extremely useful in simplifying trigonometric expressions and solving equations. Some of the most fundamental identities include the Pythagorean identities, like \(\sin^2(\theta) + \cos^2(\theta) = 1\), which express the direct relationship between the sine and cosine of the same angle.

Another useful set of identities are the angle sum and difference identities, which allow for the calculation of the sine, cosine, or tangent of an angle that is formed from the sum or difference of two known angles. For exercises like finding \(\sin 45^\circ + \cos 45^\circ\), knowing identities isn't directly needed because we're dealing with the actual values, but understanding how these identities work could greatly assist in more complex problems where the values are not as straightforward.