Verified answer

Note that sec^2(x) - tan^2(x) = 1

Multiplying the top and the bottom of 5/(tan(x)+sec(x)) leads to

5[sec(x) - tan(x)]

multiply numerator & denominator by sec(x)-tan(x). Denomintor will be sec^2(x)-tan^(x) and equal to 1. So answer is 5[sec(x)-tan(x)].

5 / (tan(x) + sec(x)) =

5 / (sin(x)/cos(x) + 1/cos(x)) =

5 / ( (sin(x) + 1) / cos(x) ) =

5 * cos(x) / (sin(x) + 1) =

5 * cos(x) * (sin(x) - 1) / (sin(x)^2 - 1) =

5 * cos(x) * (1 - sin(x)) / (1 - sin(x)^2) =

5 * cos(x) * (1 - sin(x)) / cos(x)^2 =

5 * (1 - sin(x)) / cos(x) =

5 * ( (1/cos(x)) - sin(x)/cos(x)) =

5 * (sec(x) - tan(x))

5/(tanx+secx) =

(tanx - secx)/(tanx - secx) 5/(tanx + secx) =

(5tanx - 5secx)/(tan^2(x) - sec^2(x)) =

(5tanx - 5secx)/(sec^2(x) - 1 - sec^2(x)) =

5tanx - 5secx