Question 1000033
I would do these simplifications


{{{1/(-sqrt(2)/2)=(1/1)/(-sqrt(2)/2)}}}


{{{1/(-sqrt(2)/2)=(1/1)*(-2/sqrt(2))}}}


{{{1/(-sqrt(2)/2)=-2/sqrt(2)}}}


{{{1/(-sqrt(2)/2)=(-2/sqrt(2))*(sqrt(2)/sqrt(2))}}} Multiply the fraction by {{{(sqrt(2)/sqrt(2))}}} to rationalize the denominator


{{{1/(-sqrt(2)/2)=-2*sqrt(2)/(sqrt(2)*sqrt(2))}}}


{{{1/(-sqrt(2)/2)=-2*sqrt(2)/(2)}}}


{{{1/(-sqrt(2)/2)=-sqrt(2)}}}


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So in the end, {{{sec ( 11 pi / 4 )=-sqrt(2)}}}