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)

How many shares did he have to begin with?