use a sum or difference formula to find the exact value of cos 15°\nfind the exact value of the…

Explanation:

Step1: Express (15^{\circ}) as a difference

We know that (15^{\circ}=45^{\circ}-30^{\circ}).

Step2: Use the cosine - difference formula

The cosine - difference formula is (\cos(A - B)=\cos A\cos B+\sin A\sin B). Here (A = 45^{\circ}) and (B=30^{\circ}). We know that (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\cos30^{\circ}=\frac{\sqrt{3}}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), and (\sin30^{\circ}=\frac{1}{2}). Substitute these values into the formula: [ \begin{align*} \cos(45^{\circ}-30^{\circ})&=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}\ &=\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})