Find the exact value by using a half-angle identity.
-$\cos \left( - \frac { \pi } { 8 } \right)$
A) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
B) $\frac { 1 } { 2 } \sqrt { 1 - \sqrt { 2 } }$
C) $\frac { 1 } { 2 } \sqrt { 1 + \sqrt { 2 } }$
D) $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }$

Question

Quizplus · Accepted Answer

The half-angle formula for cosine is $\cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$. For $\theta = -\frac{\pi}{4}$ (since $-\frac{\pi}{8}$ is half of $-\frac{\pi}{4}$), $\cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$. Substituting into the formula gives $\cos \left( -\frac{\pi}{8} \right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \frac{1}{2}\sqrt{2 + \sqrt{2}}$, which matches choice D.

Find the Exact Value by Using a Half-Angle Identity cos⁡(−π8)\cos \left( - \frac { \pi } { 8 } \right)cos(−8π​)

Find the Exact Value by Using a Half-Angle Identity $\cos \left( - \frac { \pi } { 8 } \right)$