**66. Find an exact expression for \(\cos \frac{\pi}{16}\).****Solution:**To find the exact expression for \(\cos \frac{\pi}{16}\), we can use the half-angle formula. The half-angle formula for cosine is:\[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]In this case, \(\theta = \frac{\pi}{8}\). Rewriting the formula, we get:\[ \cos \frac{\pi}{16} = \sqrt{\frac{1 + \cos \frac{\pi}{8}}{2}} \]Now we need to find \(\cos \frac{\pi}{8}\). To do this, we use the half-angle formula again with \(\theta = \frac{\pi}{4}\):\[ \cos \frac{\pi}{8} = \sqrt{\frac{1 + \cos \frac{\pi}{4}}{2}} \]We know that \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\). Substituting this value in:\[ \cos \frac{\pi}{8} = \sqrt{\frac{1 + \frac{1}{\sqrt{2}}}{2}} \]Let’s simplify this expression:\[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]So,\[ \cos \frac{\pi}{8} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2} \]Now substituting this into our original half-angle formula:\[ \cos \frac{\pi}{16} = \sqrt{\frac{1 + \cos \frac{\pi}{8}}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2 + \sqrt{2}}}{2}}{2}} \]Simplifying further:\[ \cos \frac{\pi}{16} = \sqrt{\frac{2 + \sqrt{2 + \sqrt{2}}}{4}} = \frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{

Name: **66. Find an exact expression for \(\cos \frac{\pi}{16}\).****Soluti
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: **66. Find an exact expression for \(\cos \frac{\pi}{16}\).****Solution:**To find the exact expression for \(\cos \frac{\pi}{16}\), we can use the half-angle formula. The half-angle formula for cosine is:\[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1