1 / (Sec x Tanx) = Sec x + tan x

Step-by-step explanation:

to prove that 1 / (sec x - tan x) = sec x + tan x

from the right hand side sec x + tan x =

from trigonometry basics

sec x = 1 / cos x

tan x = sin x / cos x

so, sec x + tan x = 1 / cosx + sinx / cosx

finding the lcm

= (1 + sinx) / cos x

= (1 + sinx) cosx / cos²x

= (1 + sinx) cosx / (1 - sinx) (1 + sinx)

(1 + sinx) cancels the (1 - sinx) at the denominator

so we have;

= cos x / 1 - sinx

1/1/cosx - sinx/cosx

remember that 1 / cos x = sec x

and also sinx/cos x = tan x

so therefore we have 1 / sec x - tanx

since LHS = RHS then we can say that 1 / sec x - tan x = sec x + tan x.