Answer :
To determine which expression is equivalent to [tex]\(\cos \left(\frac{\pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{\pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right)\)[/tex], we can use the angle sum identity in trigonometry.
The angle sum identity for cosine is given by:
[tex]\[ \cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B) \][/tex]
Here, [tex]\(A = \frac{\pi}{12}\)[/tex] and [tex]\(B = \frac{5\pi}{12}\)[/tex].
By applying the identity, we get:
[tex]\[ \cos \left(\frac{\pi}{12} - \frac{5\pi}{12}\right) = \cos \left(-\frac{4\pi}{12}\right) = \cos \left(-\frac{\pi}{3}\right) \][/tex]
Since the cosine function is even (i.e., [tex]\(\cos(-x) = \cos(x)\)[/tex]), we have:
[tex]\[ \cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) \][/tex]
Now let's check the cosine of [tex]\(\frac{\pi}{3}\)[/tex]:
[tex]\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Given the numerical results:
- [tex]\(\cos \left(\frac{\pi}{12}\right) \approx 0.9659\)[/tex]
- [tex]\(\cos \left(\frac{5 \pi}{12}\right) \approx 0.2588\)[/tex]
- [tex]\(\sin \left(\frac{\pi}{12}\right) \approx 0.2588\)[/tex]
- [tex]\(\sin \left(\frac{5 \pi}{12}\right) \approx 0.9659\)[/tex]
Substituting these into our original expression:
[tex]\[ \cos \left(\frac{\pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{\pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) \][/tex]
[tex]\[ \approx 0.9659 \times 0.2588 + 0.2588 \times 0.9659 = 0.49999999999999994 \approx 0.5 \][/tex]
This matches with [tex]\(\cos \left(\frac{\pi}{3}\right)\)[/tex], confirming that:
[tex]\[ \cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Therefore, the expression [tex]\(\cos \left(-\frac{\pi}{3}\right)\)[/tex] is equivalent to [tex]\(\cos \left(\frac{\pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{\pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right)\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{\cos \left(-\frac{\pi}{3}\right)} \][/tex]
The angle sum identity for cosine is given by:
[tex]\[ \cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B) \][/tex]
Here, [tex]\(A = \frac{\pi}{12}\)[/tex] and [tex]\(B = \frac{5\pi}{12}\)[/tex].
By applying the identity, we get:
[tex]\[ \cos \left(\frac{\pi}{12} - \frac{5\pi}{12}\right) = \cos \left(-\frac{4\pi}{12}\right) = \cos \left(-\frac{\pi}{3}\right) \][/tex]
Since the cosine function is even (i.e., [tex]\(\cos(-x) = \cos(x)\)[/tex]), we have:
[tex]\[ \cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) \][/tex]
Now let's check the cosine of [tex]\(\frac{\pi}{3}\)[/tex]:
[tex]\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Given the numerical results:
- [tex]\(\cos \left(\frac{\pi}{12}\right) \approx 0.9659\)[/tex]
- [tex]\(\cos \left(\frac{5 \pi}{12}\right) \approx 0.2588\)[/tex]
- [tex]\(\sin \left(\frac{\pi}{12}\right) \approx 0.2588\)[/tex]
- [tex]\(\sin \left(\frac{5 \pi}{12}\right) \approx 0.9659\)[/tex]
Substituting these into our original expression:
[tex]\[ \cos \left(\frac{\pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{\pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) \][/tex]
[tex]\[ \approx 0.9659 \times 0.2588 + 0.2588 \times 0.9659 = 0.49999999999999994 \approx 0.5 \][/tex]
This matches with [tex]\(\cos \left(\frac{\pi}{3}\right)\)[/tex], confirming that:
[tex]\[ \cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Therefore, the expression [tex]\(\cos \left(-\frac{\pi}{3}\right)\)[/tex] is equivalent to [tex]\(\cos \left(\frac{\pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{\pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right)\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{\cos \left(-\frac{\pi}{3}\right)} \][/tex]
