Question

find the exact value of the expression \\( \cos 255 ^ { \circ } - \cos 195 ^ { \circ } \\).

the exact value of the expression is
(simplify your answer, including any radicals. use integers or fractions for any r

Explanation:

Step1: Use the formula $\cos A-\cos B=-2\sin\frac{A + B}{2}\sin\frac{A - B}{2}$

Let $A = 255^{\circ}$ and $B=195^{\circ}$. Then $\cos255^{\circ}-\cos195^{\circ}=-2\sin\frac{255^{\circ}+195^{\circ}}{2}\sin\frac{255^{\circ}-195^{\circ}}{2}$

Step2: Calculate the values of the angles in the sine functions

First, $\frac{255^{\circ}+195^{\circ}}{2}=\frac{450^{\circ}}{2} = 225^{\circ}$, and $\frac{255^{\circ}-195^{\circ}}{2}=\frac{60^{\circ}}{2}=30^{\circ}$
So the expression becomes $-2\sin225^{\circ}\sin30^{\circ}$

Step3: Find the values of $\sin225^{\circ}$ and $\sin30^{\circ}$

We know that $\sin225^{\circ}=\sin(180^{\circ}+45^{\circ})=-\sin45^{\circ}=-\frac{\sqrt{2}}{2}$, and $\sin30^{\circ}=\frac{1}{2}$
Substitute these values into the expression: $-2\times(-\frac{\sqrt{2}}{2})\times\frac{1}{2}$

Step4: Simplify the expression

$-2\times(-\frac{\sqrt{2}}{2})\times\frac{1}{2}=\frac{\sqrt{2}}{2}$

Answer:

$\frac{\sqrt{2}}{2}$