Step-by-step explanation:

Given that secθ = x+1/4x

1+tan²θ = sec²θ

or tan²θ = sec²θ – 1

on expanding

tan²θ = ( x+1/4x)² -1

or tan²θ = ( x²+1/16x² + 1/2 -1 )

or tan²θ = (x² +1/16x² -1/2 )

or tan²θ = x² +1/16x²-1/2

or tan²θ = (x-1/4x)^2

or tan²θ = +(x-1/4x) or – (x-1/4x)

when tanθ = (x-1/4x) we get

secθ+tanθ = x+1/4x + x-1/4x = 2x

when tanθ = -(x-1/4x)

secθ+tanθ = (x+1/4x) – (x-1/4x) = 1/2x

Hence

secθ+tanθ = (x+1/4x) – (x-1/4x) = 1/2x

and secθ+tanθ = x+1/4x + x-1/4x = 2x