Certainly! Let's derive the exact value of the expression
[tex]\[ \cos \frac{6 \pi}{7} \cos \frac{29 \pi}{42}+\sin \frac{6 \pi}{7} \sin \frac{29 \pi}{42}. \][/tex]
We can use the sum-to-product identity for cosine, specifically the identity for the cosine of a sum:
[tex]\[ \cos A \cos B + \sin A \sin B = \cos (A - B). \][/tex]
In this expression, we let [tex]\( A = \frac{6 \pi}{7} \)[/tex] and [tex]\( B = \frac{29 \pi}{42} \)[/tex], and substitute these values into the identity:
[tex]\[ \cos \frac{6 \pi}{7} \cos \frac{29 \pi}{42} + \sin \frac{6 \pi}{7} \sin \frac{29 \pi}{42} = \cos \left( \frac{6 \pi}{7} - \frac{29 \pi}{42} \right). \][/tex]
Now, we need to simplify the argument of the cosine on the right-hand side:
[tex]\[ \frac{6 \pi}{7} - \frac{29 \pi}{42}. \][/tex]
To perform this subtraction, we first need a common denominator. The least common multiple of [tex]\(7\)[/tex] and [tex]\(42\)[/tex] is [tex]\(42\)[/tex]:
[tex]\[ \frac{6 \pi}{7} = \frac{6 \pi \times 6}{7 \times 6} = \frac{36 \pi}{42}. \][/tex]
Now, we subtract the fractions:
[tex]\[ \frac{36 \pi}{42} - \frac{29 \pi}{42} = \frac{36 \pi - 29 \pi}{42} = \frac{7 \pi}{42} = \frac{\pi}{6}. \][/tex]
Hence, the expression simplifies to:
[tex]\[ \cos \left( \frac{\pi}{6} \right). \][/tex]
We know from the unit circle or trigonometric values that:
[tex]\[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}. \][/tex]
Therefore, the exact value of the given expression is
[tex]\[ \cos \frac{6 \pi}{7} \cos \frac{29 \pi}{42}+\sin \frac{6 \pi}{7} \sin \frac{29 \pi}{42} = \frac{\sqrt{3}}{2}. \][/tex]
Upon cross-checking with the numerical results, we have confirmed the consistency. The final, exact value is:
[tex]\[ \boxed{\frac{\sqrt{3}}{2}}. \][/tex]
