To find the exact value of [tex]\(\cos \left(\frac{5\pi}{12}\right)\)[/tex] using the sum and difference of cosines formula, we follow these steps:
1. Express [tex]\(\frac{5\pi}{12}\)[/tex] as a sum or difference of angles with known cosine and sine values:
[tex]\[
\frac{5\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}
\][/tex]
2. Recall the cosine difference identity:
[tex]\[
\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)
\][/tex]
In this case:
[tex]\[
a = \frac{\pi}{4} \quad \text{and} \quad b = \frac{\pi}{6}
\][/tex]
3. Calculate the trigonometric values for [tex]\(a = \frac{\pi}{4}\)[/tex] and [tex]\(b = \frac{\pi}{6}\)[/tex]:
[tex]\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\][/tex]
4. Substitute these values into the cosine difference identity:
[tex]\[
\cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{6}\right)
\][/tex]
So, substituting the values:
[tex]\[
\cos\left(\frac{5\pi}{12}\right) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)
\][/tex]
5. Simplify the expression:
[tex]\[
\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2} \cdot 1}{4}
\][/tex]
[tex]\[
\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
[tex]\[
\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
Therefore, the exact value of [tex]\(\cos \left(\frac{5\pi}{12}\right)\)[/tex] is:
[tex]\[
\cos \left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
