Question 333685
<pre><font size = 4 color = "indigo"><b>

{{{"cos("}}}{{{1/2}}}{{{theta}}}{{{")"}}}{{{""=""}}}{{{""+- sqrt( (1+cos(theta))/2) }}}

Since 22.5° is {{{1/2}}} of 45°, we substitute {{{theta="45°"}}}

{{{cos("22.5°")}}}{{{""=""}}}{{{"cos("}}}{{{1/2}}}{{{""*"45°"}}}{{{")"}}}{{{""=""}}}{{{""+- sqrt( (1+cos("45°"))/2) }}}

We take the positive result because cos(22.5°) is positive.

{{{cos("22.5")}}}{{{""=""}}}{{{""+ sqrt( (1+(sqrt(2)/2)    )/2) }}}

Multiply top and bottom of the complex fraction under the radical by 2:

{{{cos("22.5")}}}{{{""=""}}}{{{sqrt( (2+sqrt(2))/4) }}}

{{{cos("22.5")}}}{{{""=""}}}{{{sqrt( 2+sqrt(2))/2 }}}

That's the final result.  We check it by evaluating that as
.9238795323 with a calculator and then finding the cosine of 22.5° 
and we find that it also has that same decimal value, .9238795323.

Edwin</pre>