#sec(theta) = 5/4#
Recall that #sec theta = 1/cos theta#. Because of this, we can simply take the reciprocal of both sides so we can work with functions we're more used to seeing.
#color(blue)(cos(theta) = 4/5)#
We are looking for #color(green)(sec(2theta))#, which can also be written in terms of trigonometric functions we are more familiar with.
#color(green)(sec(2theta))#
# = 1/cos(2theta)#
The double angle identity for cosine states that #color(red)(cos(2theta) = 2cos^2(theta) - 1)#.
# = 1/color(red)(2cos^2(theta) - 1)#
Interestingly, this means that we don't actually have to solve for #theta# to find the value of #color(green)(sec(2theta))#.
# = 1/(2(color(blue)cos(theta))^2 - 1)#
# = 1/(2(color(blue)(4/5))^2 - 1)#
# = 1/(32/25 - 1)#
# = 1/(7/25)#
# = 25/7#
#therefore# #sec(2theta) = 25/7# for #sec(theta) = 5/4#.
