find the exact value of the expression given below. \n\n\\( \\cos \\left( \\frac { 5 \\pi } { 12 } \\right)…

Explanation:

Step1: Express (\frac{5\pi}{12}) as a sum of known angles

We know that (\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}). So, (\cos(\frac{5\pi}{12})=\cos(\frac{\pi}{4}+\frac{\pi}{6}))

Step2: Use the cosine addition formula (\cos(A + B)=\cos A\cos B-\sin A\sin B)

Here (A=\frac{\pi}{4}) and (B = \frac{\pi}{6}) (\cos(\frac{\pi}{4}+\frac{\pi}{6})=\cos\frac{\pi}{4}\cos\frac{\pi}{6}-\sin\frac{\pi}{4}\sin\frac{\pi}{6}) We know that (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}), (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (\sin\frac{\pi}{6}=\frac{1}{2}) Substitute these values: [ \begin{align*} \cos(\frac{\pi}{4}+\frac{\pi}{6})&=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}\ &=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\ &=\frac{\sqrt{6}-\sqrt{2}}{4} \end{align*} ]

Answer:

(\frac{\sqrt{6}-\sqrt{2}}{4})