Question 106799
Since,
{{{cos((x/2))= sqrt(2)/2}}}
You know from the relationship,
{{{sin^2(y)+cos^2((y))=1}}}
{{{sin^2(x/2)+1/2=1}}}
{{{sin(x/2)=sqrt(2)/2}}}
As you can see on the unit circle, there are two angles (A and -A) that solve the equation,
{{{cos((x/2))= sqrt(2)/2}}}
{{{drawing( 300, 300, -1.5, 1.5, -1.5, 1.5,grid( 1 ),circle( 0, 0, 1 ),green(line(0,0,.707,-.707)),green(line( 0,0,.707,.707)),green(line(.707,.707,.707,-.707)),locate(0.3,0.2,A),locate(0.2,-0.05,-A))}}}
You can determine A several ways. 
Two ways are using the inverse trigonometric functions and geometrically.
Using inverse trig, the only angle that has equal sine and cosine is 45 degrees, or
{{{A=45^o}}}
{{{x/2=45^o}}}
{{{x=90^o}}}
{{{-A=45^o}}}
{{{x/2=-45^o}}}
{{{x=-90^o}}}
Geometrically, the triangle with A as one angle is a right tringle. It is also an right, isoceles triangle with a hypotneuse of 1 and sides of {{{sqrt(2)/2}}}
Therefore A+A=90 or A=45.