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The question asks to express each given trigonometric expression in terms of \( \cos(x) \) and \( \sin(x) \), where \( x \) is a real number.

Let's go through each expression one by one:

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### Part (a)
\[
-2 \cos(-x) + 5 \sin(-x) + 2 \cos(x + \pi) + \cos(\pi - x) + 3 \sin(\pi - x)
\]

To rewrite each term:
1. \( \cos(-x) = \cos(x) \) (cosine is an even function).
2. \( \sin(-x) = -\sin(x) \) (sine is an odd function).
3. \( \cos(x + \pi) = -\cos(x) \).
4. \( \cos(\pi - x) = -\cos(x) \).
5. \( \sin(\pi - x) = \sin(x) \).

Substitute each:
\[
-2 \cos(x) - 5 \sin(x) - 2 \cos(x) - \cos(x) + 3 \sin(x)
\]
Simplify:
\[
-5 \cos(x) - 2 \sin(x)
\]

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### Part (b)
\[
\sin(\pi - x) + \sin(-x) + 3 \cos(x + \pi)
\]

Rewrite each term:
1. \( \sin(\pi - x) = \sin(x) \).
2. \( \sin(-x) = -\sin(x) \).
3. \( \cos(x + \pi) = -\cos(x) \).

Substitute:
\[
\sin(x) - \sin(x) - 3 \cos(x)
\]
Simplify:
\[
-3 \cos(x)
\]

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### Part (c)
\[
2 \sin(4 \pi - x) + 2 \cos(5 \pi + x) - \sin(7 \pi - x)
\]

Rewrite each term:
1. \( \sin(4 \pi - x) = -\sin(x) \).
2. \( \cos(5 \pi + x) = -\cos(x) \).
3. \( \sin(7 \pi - x) = -\sin(x) \).

Substitute:
\[
2(-\sin(x)) + 2(-\cos(x)) - (-\sin(x))
\]
Simplify:
\[
-2 \sin(x) - 2 \cos(x) + \sin(x) = - \sin(x) - 2 \cos(x)
\]

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### Part (d)
\[
2 \sin(\pi - x) + \cos(x - \pi)
\]

Rewrite each term:
1. \( \sin(\pi - x) = \sin(x) \).
2. \( \cos(x - \pi) = -\cos(x) \).

Substitute:
\[
2 \sin(x) - \cos(x)
\]

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Would you like detailed steps for each term or have any questions?

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### Related Questions
1. How do even and odd properties of trigonometric functions help in simplifications?
2. Can cosine and sine functions have phase shifts that simplify to basic trigonometric functions?
3. What are the periodic properties of sine and cosine?
4. How do you find equivalent angles for trigonometric simplifications?
5. What is the significance of symmetry in trigonometric identities?

### Tip
When simplifying trigonometric expressions, always check if angles can be simplified using properties like even/odd functions or periodicity.

Soit x un réel, exprimer les expressions suivantes en fonction de cos(x) et de sin(x):

a. -2 cos(-x) + 5 sin(-x) + 2 cos(x + π) + cos(π - x) + 3 sin(π - x)
b. sin(π - x) + sin(-x) + 3 cos(x + π)
c. 2 sin(4π - x) + 2 cos(5π + x) - sin(7π - x)
d. 2 sin(π - x) + cos(x - π)

Learn how to express trigonometric expressions in terms of cos(x) and sin(x) using even-odd properties and angle transformations. This example simplifies expressions with mixed angle terms, suitable for students in Grades 11-12.

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