The expression given is \(\sin(x) - \cos(x)\). This trigonometric expression can be simplified or evaluated depending on the context. Below are a few things you can do with this expression:

### 1. Simplification
The expression \(\sin(x) - \cos(x)\) can be simplified using the identity:
\[
\sin(x) - \cos(x) = \sqrt{2} \sin\left(x - \frac{\pi}{4}\right)
\]

### 2. Expression in Terms of a Single Trigonometric Function
You can express \(\sin(x) - \cos(x)\) in terms of a single trigonometric function:
\[
\sin(x) - \cos(x) = \sqrt{2} \cos\left(x + \frac{\pi}{4}\right)
\]

### 3. Maximum and Minimum Values
The expression \(\sin(x) - \cos(x)\) can take values between \(-\sqrt{2}\) and \(\sqrt{2}\).

### 4. Solving Equations Involving \(\sin(x) - \cos(x)\)
If you need to solve an equation like \(\sin(x) - \cos(x) = 0\), you would find the general solutions:
\[
\sin(x) = \cos(x)
\]
This gives:
\[
x = \frac{\pi}{4} + n\pi, \text{ where } n \text{ is any integer}
\]

### 5. Derivatives and Integrals
If you need the derivative or integral of \(\sin(x) - \cos(x)\):
- **Derivative**: \(\cos(x) + \sin(x)\)
- **Integral**: \(-\cos(x) - \sin(x) + C\), where \(C\) is the constant of integration.

Would you like further details on any of these points, or do you have any specific questions?

Here are 5 questions related to this topic:
1. What is the graphical representation of \(\sin(x) - \cos(x)\)?
2. How do you derive the maximum and minimum values of \(\sin(x) - \cos(x)\)?
3. What are the applications of the expression \(\sin(x) - \cos(x)\) in physics or engineering?
4. How can we express \(\sin(x) - \cos(x)\) using Euler's formula?
5. How does the phase shift in the identity \(\sin(x) - \cos(x) = \sqrt{2} \sin\left(x - \frac{\pi}{4}\right)\) affect the graph of the function?

**Tip**: Recognizing and using trigonometric identities can simplify complex expressions and solve equations more easily.

Explore how to simplify the trigonometric expression sin(x) - cos(x) using identities and solve related equations. Learn about its derivative, integral, and the maximum and minimum values of this expression. Suitable for Grades 10-12 students interested in trigonometry.

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