Question

find the exact value of the expression

cos 255$°$ - cos 195$°$

Short answer

cos255$°$-cos195$°==0.7071067812$

Step-by-step solution

Step 1. Rewriting the expression  

$$\begin{gathered} \cos (270°-x)=-\sin x \\ \cos (180°+x)=-\cos x \\ \cos 255°=\cos (270°-15°)=-\sin 15° \\ \cos 195°=\cos (180°+15°)=-\cos 15° \end{gathered}$$

Step 2. Calculations 

Now our expression becomes

$$\begin{gathered} =\cos 255°-\cos 195=\cos 15°-\sin 15° \\ Nowweknowthat \\ \cos 15°=\sin 75° \\ So, \\ \cos 15°-\sin 15°=\sin 75°-\sin 15° \\ \sin x-\sin y=2\sin \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right) \\ So,\sin 75°-\sin 15°=2(\sin 30°\cos 45°)=2\frac{1}{2}\frac{1}{\sqrt{2}} \\ =0.7071067812 \end{gathered}$$