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

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}). By the cosine - of - a - sum formula (\cos(A + B)=\cos A\cos B-\sin A\sin B), where (A=\frac{\pi}{4}) and (B = \frac{\pi}{6}).

Step2: Substitute the values of (\cos A), (\sin A), (\cos B), and (\sin B)

We know that (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}), and (\sin\frac{\pi}{6}=\frac{1}{2}). Substitute these values into the formula: [ \begin{align*} \cos\left(\frac{\pi}{4}+\frac{\pi}{6}\right)&=\cos\frac{\pi}{4}\cos\frac{\pi}{6}-\sin\frac{\pi}{4}\sin\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})