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The cosine difference formula is a key identity in trigonometry. This formula is used when you need to find the cosine of the difference between two angles. The formula is as follows:

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

When using this formula, it helps in breaking down complex trigonometric expressions into simpler components.
For example, in the original exercise, the expression \( \cos(\frac{5\pi}{4} - x) \) can be expanded using this formula.
You substitute \( a = \frac{5\pi}{4} \) and \( b = x \).
This turns \( \cos(\frac{5\pi}{4} - x) \) into \( \cos(\frac{5\pi}{4})\cos x + \sin(\frac{5\pi}{4})\sin x\).
Understanding and applying the cosine difference formula makes tackling such trigonometric identities much more manageable.

Understanding the values of cosine and sine for specific angles is crucial. These values often appear in various trigonometric identities and problems.
For common angles, like \( \frac{\pi}{4}, \frac{3\pi}{4}, \text{or} \frac{5\pi}{4} \), these are typically memorized to make solving problems easier.
In the exercise, we deal with \( \frac{5\pi}{4} \), which is an angle in the third quadrant of the unit circle.

• \( \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
• \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)

These values are derived from the reference angle \( \frac{\pi}{4} \) and the properties of the unit circle.
Using these known values, you can substitute them into your trigonometric expression to simplify it further. This substitution is what leads to the next step of proving the identity.

Trigonometric simplification involves reducing a complex trigonometric expression into a simpler form. This is essential when proving identities.
In our exercise, after substituting the cosine and sine values, the expression becomes:
\( -\frac{\sqrt{2}}{2}\cos x -\frac{\sqrt{2}}{2}\sin x \).
The goal is to express it in the simplest form possible to match the right side of the identity.
You can factor out the common factor \( -\frac{\sqrt{2}}{2} \) from both terms:

• \( -\frac{\sqrt{2}}{2}(\cos x + \sin x) \)

This process of factoring simplifies the expression, showing it is equal to \( -\frac{\sqrt{2}}{2}(\cos x + \sin x) \), hence proving the original identity.Understanding how to factor and simplify correctly is a key skill in trigonometry.
This skill helps in making complicated problems easier to handle and solves them efficiently.