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.