sec(x) csc(x)/1 tan(x)=csc(x)

********that is 1+tan(x) on the bottom my typo********

Use \sin x=\cos x\tan x :

\frac{(\frac{1}{\cos x}+\frac{1}{\sin x})}{(1+\tan x)}
\frac{\sin x+\cos x}{\cos x\sin x}\frac{1}{1+\tan x}=
\frac{\sin x+\cos x}{\cos x\sin x + \sin^2 x}=
\frac{\sin x+\cos x}{\sin x(\cos x+\sin x)}=\frac{1}{\cos x}

Oh, a typo at the last equal sign: it is obviously \frac{1}{\sin x} .