sec x csc x/tan x cot x= sin x cos x

flat out not true. Counter-example: let x=pi/4.
Then sin x = cos x = 1/√2, sec x = csc x = √2, and tan x =cot x =1.
So sec x csc x/tanx cot x = 2, but sin x cos x = 1/2.

didn't come out right.
x = \frac{\pi}{4}
\sin x = \cos x = \frac{1}{\sqrt{2}}, \sec x = \csc x = \sqrt{2}, and \tan x = \cot x =1.
So sec x csc x/tanx cot x = 2, but sin x cos x = 1/2.

I guess some plus sign is missing, which is common on this site unless "math" environment is used.

I think what Javaris2011 intended was

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

Too long, the last equal sign is followed by \cos x+\sin x, not only \cos x