To find the exact value of [tex]\(\cos \frac{5\pi}{12}\)[/tex], we will use the cosine addition formula:
[tex]\[
\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)
\][/tex]
We need to express [tex]\(\frac{5\pi}{12}\)[/tex] as a sum of two angles whose trigonometric functions are known. A good choice is to split it into [tex]\(\frac{\pi}{3} + \frac{\pi}{4}\)[/tex]:
[tex]\[
\cos \frac{5\pi}{12} = \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right)
\][/tex]
Now, apply the cosine addition formula:
[tex]\[
\cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \cos\frac{\pi}{3}\cos\frac{\pi}{4} - \sin\frac{\pi}{3}\sin\frac{\pi}{4}
\][/tex]
We know the following values for [tex]\(\frac{\pi}{3}\)[/tex] and [tex]\(\frac{\pi}{4}\)[/tex]:
[tex]\[
\cos\frac{\pi}{3} = \frac{1}{2}, \quad \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, \quad \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}
\][/tex]
Substitute these values into the formula:
[tex]\[
\cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
\][/tex]
Calculate each term:
[tex]\[
\cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}
\][/tex]
Combine the terms:
[tex]\[
\cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{2} - \sqrt{6}}{4}
\][/tex]
This simplifies to:
[tex]\[
\cos \frac{5\pi}{12} = \frac{\sqrt{2}(\sqrt{3} - 1)}{4}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{2}(\sqrt{3} - 1)}{4}}
\][/tex]
So the correct choice is D: [tex]\(\frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)[/tex].
