find the exact value of the expression given below.\ncos (105°)\n\ncos (105°)=□\n(simplify your answer…

Explanation:

Step1: Use the cosine addition formula

$$\cos(A + B)=\cos A\cos B-\sin A\sin B$$ Let (A = 60^{\circ}) and (B = 45^{\circ}), then (\cos(105^{\circ})=\cos(60^{\circ}+ 45^{\circ}))

Step2: Substitute the values of trigonometric functions

We know that (\cos60^{\circ}=\frac{1}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin60^{\circ}=\frac{\sqrt{3}}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}) [ \begin{align*} \cos(60^{\circ}+45^{\circ})&=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}\ &=\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})