Question

How do I know that the following equation is true:

Answer 1

Answer:

Half angle Identity for cosine:

$\cos \left(\frac{\theta}{2} \right)=\pm \sqrt{\frac{1+\cos(\theta)}{2} }$

$\text{Note that it is} \pm \text{because we don't know which Quadrant the half-angle is, }$$\text{the information of}\cos(\theta) \text{is not enough to conclude the angle}$

$\frac{\pi}{8} \text{ is a Quadrant I angle, therefore, positive}$

$\cos \left(\frac{\pi}{8} \right)=\sqrt{\frac{1+\cos(\frac{\pi}{4} )}{2} }$

The equation is true.