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Co-function identities are very important in trigonometry as they show the relationship between different trigonometric functions. Specifically, they display how one function can be expressed in terms of another, shifting the angle. For sine and cosine, these identities are especially helpful:

• \(\sin(90^{\circ} - x) = \cos(x)\)
• \(\cos(90^{\circ} - x) = \sin(x)\)

In the problem we're solving, using the co-function identity, \(\sin(90^{\circ} + x)\) converts to \(\cos(x)\). This shows how angles can switch the role of sine and cosine when part of a right angle addition or subtraction. By understanding these identities, you're better equipped to simplify or prove identities like the one given in this exercise.

The sine and cosine relationship is central to understanding trigonometric concepts. They're known as co-functions due to their close link through co-function identities. This means that if you have the sine of an angle, you can easily find the cosine of the complementary angle.For example, \(\sin(90^{\circ} - x) = \cos(x)\), and \(\cos(90^{\circ} - x) = \sin(x)\). These relationships help to simplify many trigonometric expressions and identities. They reflect the geometric relationship between angles in a right triangle or on the unit circle. Whenever you manipulate expressions using sine and cosine, remember how their roles can interchange when involving right-angle complements.

The angle addition and subtraction identities help to determine the trigonometric function values for angles formed by adding or subtracting known angles. The formula for sine angle addition is:

• \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\)

And for cosine:

• \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\)

These identities are fundamentals for evaluating trigonometric functions of non-standard angles.In this exercise, you can see how adding or subtracting angles in terms like \(90^{\circ}\) flips the trigonometric function through co-function identities. Knowing these relationships and identities allows you to break down complex trigonometric expressions into simpler ones, making problems and their solutions much clearer.