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When you see an expression like \( \cos(a - b) \), it matches a specific trigonometric identity known as the cosine difference formula. This formula is super handy for simplifying such expressions. The cosine difference formula states that: \[ \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b) \]By recognizing this form, you're instantly on track to simplifying the problem. For our current problem, \( \cos \left(\frac{3\pi}{4} - x\right) \), it fits perfectly in the formula where \( a = \frac{3\pi}{4} \) and \( b = x \). Plugging in these values takes us one step closer to the solution.

Knowing exact trigonometric values for key angles can save you a lot of time. For angles like \( \frac{3\pi}{4} \), you don't need to calculate every time. Memorize some key values:

• \(\cos \frac{3\pi}{4} = -\frac{1}{\sqrt{2}} \)
• \(\sin \frac{3\pi}{4} = \frac{1}{\sqrt{2}} \)

These come from understanding the unit circle and knowing in which quadrant the angle falls. For \( \frac{3\pi}{4} \), it is in the second quadrant where sine is positive and cosine is negative. Plugging these values back into our equation makes everything much simpler.

Simplifying compared to the original form is one of the last steps. Once we substitute the exact trigonometric values into our formula, we get: \[ \cos \left(\frac{3\pi}{4} - x\right) = -\frac{1}{\sqrt{2}} \cos(x) + \frac{1}{\sqrt{2}} \sin(x) \]
To make it look even cleaner, we often factor common terms. Here, \(\frac{1}{\sqrt{2}}\) is the common term. Factoring it out helps simplify: \[ \cos \left(\frac{3\pi}{4} - x\right) = \frac{1}{\sqrt{2}} \left(-\cos(x) + \sin(x) \right) \]
Seeing the final form should give you a sense of achievement. Every step you took -- from recognizing the formula to using exact values and finally simplifying -- all came together in a clean and simple form!