Identities:
#color(red)bb(secx=1/cosx) \ \ \ \ \[1]#
#color(red)bb(cos(A+B)=cosAcosB-sinAsinB) \ \ \ \[2]#
#cos((5pi)/4)=cos(pi+pi/4)#
Using #[2]#
#cos(pi+pi/4)=cos(pi)cos(pi/4)-sin(pi)sin(pi/4)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(-1)(sqrt(2)/2)-(0)((sqrt(2))/2)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-sqrt(2)/2#
Using #[1]#
#1/cosx\ \ \ \ \ \ \ \ \ \ \ \ =-1/(sqrt(2)/2)=-sqrt(2)#
#:.#
#color(blue)(sec((5pi)/4)=-sqrt(2))#