To find the exact value of the expression [tex]\(\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)\)[/tex], let's break down the steps and evaluate each component individually.
First, recall the exact values of the trigonometric functions at specific angles:
1. [tex]\(\cos(\pi)\)[/tex]:
[tex]\[
\cos(\pi) = -1
\][/tex]
2. [tex]\(\cos\left(\frac{\pi}{3}\right)\)[/tex]:
[tex]\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\][/tex]
3. [tex]\(\sin(\pi)\)[/tex]:
[tex]\[
\sin(\pi) = 0
\][/tex]
4. [tex]\(\sin\left(\frac{\pi}{3}\right)\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Next, substitute these values into the expression:
[tex]\[
\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)
\][/tex]
We need to evaluate each part of the expression:
[tex]\[
\cos (\pi) \cos \left(\frac{\pi}{3}\right) = (-1) \cdot \left(\frac{1}{2}\right) = -\frac{1}{2}
\][/tex]
[tex]\[
\sin (\pi) \sin \left(\frac{\pi}{3}\right) = 0 \cdot \left(\frac{\sqrt{3}}{2}\right) = 0
\][/tex]
Now combine these results:
[tex]\[
-\frac{1}{2} - 0 = -\frac{1}{2}
\][/tex]
Hence, the exact value of the expression [tex]\(\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
So, the correct answer is:
[tex]\[
-\frac{1}{2}
\][/tex]
