I need to prove this identity: cos(x)/1-sin(x)=sec(x)+tan(x).?

Answer 1

LHS
cos x / (1 - sin x)
1 / [ (1 - sin x) / cos x ]
1 / [ sec x - tan x ]
(sec x + tan x) / (sec x - tan x)(sec x + tan x)
(sec x + tan x) / (sec ² x - tan ² x)
(sec x + tan x / (tan ² x + 1 - tan ² x)
sec x + tan x

RHS
sec x + tan x

LHS = RHS

Answer 2

cosx / (1 - sinx)

Multiply 1 + sinx for both top and bottom

(cosx + sinx cosx) / (1 - sin^2(x))

trig identity
sin^2(x) = 1 - cos^2(x)

substitute

(cosx + sinx cosx) / [1 - (1 - cos^2(x))]

simplify
(cosx + sinx cosx) / (cos^2(x))

this becomes:
cosx / cos^2(x) + (sinx cosx) / (cos^2(x))

simplify
1/cosx + sinx/cosx

sec(x) + tan(x)


hope it helps.


Rec

Answer 3

sec(x) + tan(x) = 1/cos(x) + sin(x)/cos(x)
= 1+sin(x)
-----------------
cos(x)

Now multiply and divide by (1-sin(x))

= (1+sin(x))*(1-sin(x))
-------------------------------
cos(x) * (1-sin(x))

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

= cos(x) * cos(x)
------------------------ (since cos(x)*cos(x) + sin(x)*sin(x) = 1)
cos(x) * (1-sin(x))

= cos(x)
---------------
1-sin(x)

and that's the proof................

cheers