Let's solve the given equation step-by-step:
[tex]\[
\cos \frac{\pi}{4} \cos \frac{\pi}{6} = \frac{1}{2} \left( -\sin \frac{\pi}{12} + \cos \frac{5\pi}{12} \right)
\][/tex]
1. Calculate [tex]\(\cos \frac{\pi}{4}\)[/tex]:
Recall that [tex]\(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)[/tex].
From calculations:
[tex]\[
\cos \frac{\pi}{4} = 0.7071067811865476
\][/tex]
2. Calculate [tex]\(\cos \frac{\pi}{6}\)[/tex]:
Recall that [tex]\(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)[/tex].
From calculations:
[tex]\[
\cos \frac{\pi}{6} = 0.8660254037844387
\][/tex]
3. Calculate the left-hand side of the equation:
[tex]\[
\cos \frac{\pi}{4} \cdot \cos \frac{\pi}{6} = 0.7071067811865476 \cdot 0.8660254037844387
\][/tex]
This yields:
[tex]\[
\cos \frac{\pi}{4} \cos \frac{\pi}{6} = 0.6123724356957946
\][/tex]
4. Calculate [tex]\(\sin \frac{\pi}{12}\)[/tex]:
From calculations:
[tex]\[
\sin \frac{\pi}{12} = 0.25881904510252074
\][/tex]
5. Calculate [tex]\(\cos \frac{5 \pi}{12}\)[/tex]:
From calculations:
[tex]\[
\cos \frac{5 \pi}{12} = 0.25881904510252074
\][/tex]
6. Calculate the right-hand side of the equation:
[tex]\[
-\sin \frac{\pi}{12} + \cos \frac{5 \pi}{12} = -0.25881904510252074 + 0.25881904510252074 = 0
\][/tex]
Therefore:
[tex]\[
\frac{1}{2} \left( -\sin \frac{\pi}{12} + \cos \frac{5\pi}{12} \right) = \frac{1}{2} \cdot 0 = 0
\][/tex]
7. Compare Left-Hand Side (LHS) and Right-Hand Side (RHS):
From calculations:
- LHS = [tex]\(0.6123724356957946\)[/tex]
- RHS = [tex]\(0.0\)[/tex]
Thus, the values confirm that the given equation does not hold true as the left-hand side of [tex]\(0.6123724356957946\)[/tex] is not equal to the right-hand side of [tex]\(0.0\)[/tex].
