Question

if I assume that
cos(x)+5sin(x)=23/5
and
2cos(x)+sin(x)=10/5
how would I find the exact numeric value of sec(x) and tan(x)?

Answer 1

think of cos(x) and sin(x) as two unknowns - call them 'c' and 's' for example, so we have:
c = cos(x)
s = sin(x)

then write the equations using these new symbols as follows:
c + 5s = 23/5
2c + s = 10/5

you then have two equations with two unknowns which you should be able to solve to find c and s.

then you can use these to get the required values as follows:
sec(x) = 1/cos(x) = 1/c
tan(x) = sin(x)/cos(x) = s/c