To determine whether the statement [tex]\(\cos(-x) = -\cos(x)\)[/tex] is true for all values of [tex]\(x\)[/tex], let's consider the nature of the cosine function.
The cosine function, [tex]\(\cos(x)\)[/tex], has a specific property known as even symmetry. This property means that the cosine of a negative angle is equal to the cosine of the corresponding positive angle:
[tex]\[
\cos(-x) = \cos(x)
\][/tex]
To verify this, let's check the identity at specific values of [tex]\(x\)[/tex]:
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[
\cos(-0) = \cos(0)
\][/tex]
[tex]\[
1 = 1
\][/tex]
This matches [tex]\(\cos(0) = \cos(0)\)[/tex], not [tex]\(-\cos(0)\)[/tex].
2. When [tex]\( x = \frac{\pi}{2} \)[/tex] (90 degrees):
[tex]\[
\cos\left(-\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right)
\][/tex]
[tex]\[
0 = 0
\][/tex]
This matches [tex]\(\cos\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right)\)[/tex], not [tex]\(-\cos\left(\frac{\pi}{2}\right)\)[/tex].
3. When [tex]\( x = \pi \)[/tex] (180 degrees):
[tex]\[
\cos(-\pi) = \cos(\pi)
\][/tex]
[tex]\[
-1 = -1
\][/tex]
This matches [tex]\(\cos(\pi) = \cos(\pi)\)[/tex], not [tex]\(-\cos(\pi)\)[/tex].
From this analysis of specific values, we see that the cosine of [tex]\( -x \)[/tex] equals the cosine of [tex]\( x \)[/tex], not the negative of it. Thus, the statement [tex]\(\cos(-x) = -\cos(x)\)[/tex] is false for all values of [tex]\(x\)[/tex].
Hence, the correct answer is:
B. False
