find the exact value of the expression given below.\n\ncos (-\\frac {7\\pi}{12})\n\ncos (-\\frac…

Explanation:

Step1: Use the property of cosine function

Since (\cos(-x)=\cos(x)), so (\cos\left(-\frac{7\pi}{12}\right)=\cos\left(\frac{7\pi}{12}\right)). And (\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}).

Step2: Apply the cosine addition formula

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

Answer:

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