Recall the definition of a function #sec(phi)#:
#sec(phi)=1/cos(phi)#
Recall the definition a function #cos(phi)#:
#cos(phi)# is an abscissa (X-coordinate) of an endpoint of a radius-vector in the unit circle that forms an angle #phi# with a positive direction of the X-axis, counting counter-clockwise from the positive direction of the X-coordinate towards a radius-vector.
From this definition of a function #cos(phi)# follows that
(a) Function #cos(phi)# is periodical with a period of #2pi#.
(b) Function #cos(phi)# is even in terms of #cos(phi)=cos(-phi)#.
Using these properties, we can state that
#cos((13pi)/4)=cos((13pi)/4-4pi)=cos(-(3pi)/4)=cos((3pi)/4)#
The angle #(3pi)/4# lies in the second quarter and the abscissa of an endpoint of a unit vector that corresponds to this angle is negative.
The corresponding angle with positive but equal by absolute value abscissa is, obviously, #pi-(3pi)/4=pi/4#.
So, we can conclude that #cos((13pi)/4)=-cos(pi/4)=-sqrt(2)/2#.
Now we can calculate the value of #sec((13pi)/4)#:
#sec((13pi)/4) = 1/cos((13pi)/4) = 1/[-sqrt(2)/2]=-2/sqrt(2)=-sqrt(2)#
So, the answer is
#sec((13pi)/4) = -sqrt(2)#
