Question

Prove the identity: [(tanx)/(1-cosx) ] = [(cscx)(1+secx)]

Answer 1

tan x / ( 1 - cos x ) =
$= \frac{ \frac{sinx}{cosx} }{1-cos x}= \\ = \frac{sin x}{( 1 - cos x ) cos x}* \frac{1+cosx}{1+cos x}= \\ =\frac{sin x(1+cos x)}{sin^{2} xcos x}= \\ =\frac{1+cos x}{sin x cos x}= \frac{1}{sin x}* \frac{1+cos x}{cos x}= \\ =csc x *(1+ \frac{1}{cos x})=$
= csc x * ( 1 + sec x )
( correct )