Prove The identity:

SecX-TanX=1-SinX/CosX ... ?

Secx + tanx = cosx / (1-sinx)

Taking LHS

= Secx + Tanx

sec x = 1/cos x and

tan x = sin x / cox x

Putting the values of secx and tanx you will get

= secx + tanx

= 1/cosx + sinx / cosx

By Taking LCM you will get

= (1 + sinx) / cosx

Multiply the numerator and denominator by (1-sinx)

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

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

As 1-sin^2 x = cos^2 x therefore;

= cos^2 x / [cosx (1-sin x) ]

Divided numerator and denominator by cos x you will get

= cosx / (1-sin x)