Use a half-angle formula to find the exact value of cos(7pi )/(8) cos(7pi )/(8)= square

To find the exact value of \(\cos \frac{7\pi}{8}\) using a half-angle formula, we first need to express \(\frac{7\pi}{8}\) in a form that allows us to apply the half-angle formula. <br /><br />The half-angle formula for cosine is:<br />\[<br />\cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}<br />\]<br /><br />We need to express \(\frac{7\pi}{8}\) as \(\frac{\theta}{2}\). Let \(\theta = \frac{7\pi}{4}\). Then:<br />\[<br />\frac{7\pi}{8} = \frac{1}{2} \cdot \frac{7\pi}{4}<br />\]<br /><br />Now, we can use the half-angle formula:<br />\[<br />\cos \frac{7\pi}{8} = \cos \left( \frac{1}{2} \cdot \frac{7\pi}{4} \right) = \pm \sqrt{\frac{1 + \cos \frac{7\pi}{4}}{2}}<br />\]<br /><br />Next, we need to find \(\cos \frac{7\pi}{4}\). Since \(\frac{7\pi}{4}\) is in the fourth quadrant, where cosine is positive, and \(\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}\), we have:<br />\[<br />\cos \frac{7\pi}{4} = \cos \left( 2\pi - \frac{\pi}{4} \right) = \cos \left( -\frac{\pi}{4} \right) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}<br />\]<br /><br />Substituting this value into the half-angle formula:<br />\[<br />\cos \frac{7\pi}{8} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}<br />\]<br /><br />Simplify the expression inside the square root:<br />\[<br />\cos \frac{7\pi}{8} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} = \sqrt{\frac{2 + \sqrt{2}}{2}} = \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2}}<br />\]<br /><br />Rationalizing the denominator:<br />\[<br />\cos \frac{7\pi}{8} = \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2}} = \frac{\sqrt{2 + \sqrt{2}} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2(2 + \sqrt{2})}}{2} = \frac{\sqrt{4 + 2\sqrt{2}}}{2}<br />\]<br /><br />Thus, the exact value of \(\cos \frac{7\pi}{8}\) is:<br />\[<br />\cos \frac{7\pi}{8} = \frac{\sqrt{4 + 2\sqrt{2}}}{2}<br />\]