Let #theta = cos^-1(sqrt2/2)#

#costheta=sqrt2/2#

So in an imaginary right-angled triangle, the length of the #"hyp"# is #2# and the length of the #"adj"# is #sqrt2#. This means that the length of the #"opp"# is #sqrt(2^2-(sqrt2)^2)=sqrt2#.

Since the #"adj"# and #"opp"# are equal lengths, our triangle is isosceles. This means that it also has two angles of equal lengths. Since one angle is #pi/2#, the other two must be #pi/4#.

So if we say that #theta=pi/4#, then #cos(pi/4)="adj"/"hyp"=sqrt2/2#

#thereforepi/4=cos^-1(sqrt2/2)#