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In trigonometry, the cosine difference identity is an essential formula used to express the cosine of the difference between two angles in terms of the sines and cosines of those angles separately. This identity states:

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

This formula is particularly useful because it allows you to break down complex expressions into simpler components that are more manageable. When you learn this identity, you're gaining a tool that can make solving trigonometric equations much easier. For instance, in the exercise where you need to verify \(\cos \left(x-\frac{\pi}{4}\right)\), recognize that \(a = x\) and \(b = \frac{\pi}{4}\). By applying this identity, you can rewrite the expression and simplify your calculations. This identity is foundational in trigonometry and critical for solving diverse problems."

The angle subtraction formula is a tool used to derive values of trigonometric functions for angles that are not typically found on standard trigonometric tables. By subtracting known angles, we can calculate the trigonometric values for tougher angles. The angle subtraction formula for cosine is particularly interesting as it utilizes both sine and cosine functions:

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

This formula is versatile. By knowing the individual cosine and sine of specific angles like \(\frac{\pi}{4}\), you can derive the cosine for angles in terms of \(x\). It's essentially a strategic mathematical tool that converts problems involving angle differences into more straightforward calculations.
For example, with \(\frac{\pi}{4}\), which has the same cosine and sine value of \(\frac{\sqrt{2}}{2}\), the formulas become simpler and assist in verifying complex identities.

Trigonometric verification involves proving that two different trigonometric expressions are equivalent. To do this, you use known identities like the cosine difference identity or the angle subtraction formula to manipulate one side of the equation into becoming the other side. This process not only reinforces your understanding of the trigonometric concepts and identities but also improves problem-solving skills.
In the case of verifying \(\cos \left(x-\frac{\pi}{4}\right)\), we apply our identities and simplify step by step:

• First, use the difference identity \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).
• Substitute \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
• Simplify to match it with the given expression \(\frac{\sqrt{2}}{2}(\cos x + \sin x)\).

By practicing trigonometric verification, students gain a deeper understanding of how trigonometric identities are connected and how they can be applied to solve real problems effectively.