Here are 2 ways you could do this.
Method 1: ...almost feels like cheating ;)
If #tan(2b)=119/120#
use your calculator (or a spreadsheet) to evaluate:
#color(white)("XXX")2b=arctan(119/120)~~0.7812140874#
which implies (after dividing by 2)
#color(white)("XXX")b=0.3906070437#
Then
use your calculator again to find
#color(white)("XXX")cos(b)=cos(0.3906070437)~~0.9246780985#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Method 2: possibly more virtuous (???)
If #tan(2b)=119/120#
then we can think of angle #(2b)# as being the angle of a right triangle in standard position with the horizontal (#x#) component equal to #120# and the vertical (#y#) component equal to #119#.
The hypotenuse would be equal to #sqrt(120^2+119^2)=169#
(and, yes, I did use a calculator to discover this).
#cos(2b)=("horizontal")/("hypotenuse")=120/169#
Then, if we remember the double angle formula:
#color(white)("XXX")cos(2b)=2cos^2(b)-1#
we can re-arrange the terms to get
#color(white)("XXX")cos(b)=sqrt((cos(2b)+1)/2)#
#color(white)("XXXXXX")=sqrt((120/169+1)/2)#
#color(white)("XXXXXX")=sqrt((289/169)/2)#
#color(white)("XXXXXX")=sqrt(289/338)#
and with the aid of my calculator (again)
#color(white)("XXXXXX")~~0.924680985#
