The task here is to express each trigonometric expression in terms of $\cos(x)$ and $\sin(x)$. I'll go through each of the expressions one by one, using trigonometric identities to rewrite them.

Let's proceed with the expressions:

Expression (a):

$-2 \cos(-x) + 5 \sin(-x) + 2 \cos(x + \pi) + 11 \cos(\pi - x) + 3 \sin(x + \pi)$

Using identities:

1. $\cos(-x) = \cos(x)$.
2. $\sin(-x) = -\sin(x)$.
3. $\cos(x + \pi) = -\cos(x)$.
4. $\cos(\pi - x) = -\cos(x)$.
5. $\sin(x + \pi) = -\sin(x)$.

Substitute these identities into the expression: $-2 \cos(x) - 5 \sin(x) - 2 \cos(x) - 11 \cos(x) - 3 \sin(x)$

Combine terms: $(-2 - 2 - 11) \cos(x) + (-5 - 3) \sin(x) = -15 \cos(x) - 8 \sin(x)$

So, the answer for (a) is: $-15 \cos(x) - 8 \sin(x)$

Expression (b):

$\sin(\pi - x) + 2 \sin(-x) + \cos(x + \pi)$

Using identities:

1. $\sin(\pi - x) = \sin(x)$.
2. $\sin(-x) = -\sin(x)$.
3. $\cos(x + \pi) = -\cos(x)$.

Substitute these identities: $\sin(x) - 2 \sin(x) - \cos(x)$

Combine terms: $(1 - 2) \sin(x) - \cos(x) = -\sin(x) - \cos(x)$

So, the answer for (b) is: $-\sin(x) - \cos(x)$

Expression (c):

$2 \sin(4 \pi - x) - \cos(5 \pi + x) - \sin(7 \pi - x)$

Using identities:

1. $\sin(4 \pi - x) = -\sin(x)$ (since $\sin(2n\pi - x) = -\sin(x)$).
2. $\cos(5 \pi + x) = -\cos(x)$ (since $\cos((2n+1)\pi + x) = -\cos(x)$).
3. $\sin(7 \pi - x) = \sin(x)$ (since $\sin((2n+1)\pi - x) = \sin(x)$).

Substitute these identities: $2(-\sin(x)) - (-\cos(x)) - \sin(x)$

Simplify: $-2 \sin(x) + \cos(x) - \sin(x) = \cos(x) - 3 \sin(x)$

So, the answer for (c) is: $\cos(x) - 3 \sin(x)$

Expression (d):

$2 \sin(\pi - x) + \cos(x - \pi) + \sin(\pi + x) - \sin(x - \pi)$

Using identities:

1. $\sin(\pi - x) = \sin(x)$.
2. $\cos(x - \pi) = -\cos(x)$.
3. $\sin(\pi + x) = -\sin(x)$.
4. $\sin(x - \pi) = -\sin(x)$.

Substitute these identities: $2 \sin(x) - \cos(x) - \sin(x) + \sin(x)$

Combine terms: $(2 - 1 + 1) \sin(x) - \cos(x) = 2 \sin(x) - \cos(x)$

So, the answer for (d) is: $2 \sin(x) - \cos(x)$

Would you like further details on any of these steps?

Related Questions

1. How can we simplify trigonometric expressions involving angle addition formulas?
2. Why does $\cos(\pi - x) = -\cos(x)$ hold for all real $x$?
3. What are common identities for simplifying expressions with $\pi$-related transformations?
4. How would the answer change if we used different trigonometric identities?
5. Can these types of problems be solved graphically to verify the solutions?

Tip

When simplifying trigonometric expressions, always look for symmetries like $\pi$-shifts or negatives, which help quickly transform the terms.