Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$
A) $- \frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }$
B) $\frac { \sqrt { 2 - \sqrt { 2 } } } { 2 }$
C) $- \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 }$
D) $\frac { \sqrt { 3 - \sqrt { 3 } } } { 2 }$
E) $\frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }$

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 $\cos(22.5^\circ)$, we use $\theta = 45^\circ$, where $\cos(45^\circ) = \frac{\sqrt{2}}{2}$. Substituting into the formula gives $\cos(22.5^\circ) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \frac{\sqrt{2 + \sqrt{2}}}{2}$.

Use the Half-Angle Formulas to Determine the Exact Value of the Following