algebra

(csc(x)*cos(x)) divided by tan(x)+cot(x)  =

[1/sinx) *cos(x)]  /  [ sinx/cosx  + cosx/sinx]  =

[  cot x  ]  /    [ (sin^2 x  + cos^2 x )  /  sinx cosx ] =

[ cot x ] / [ 1 / sinx cosx ] =

[ cot x] [ sin x cos x ]  =

[cosx / sin x ] [sin x cos x ]  =

cos^2 (x)

$\frac{\csc x\cos x}{\tan x + \cot x} \\~\\ =\frac{\frac{1}{\sin x}*\cos x}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} \\~\\ =\frac{\cos x}{\sin x}*\frac{1}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} \\~\\ =\frac{\cos x}{\sin x*\frac{\sin x}{\cos x} + \sin x* \frac{\cos x}{\sin x}} \\~\\ =\frac{\cos x}{\frac{\sin^2 x}{\cos x} + \cos x} \\~\\ =\frac{\cos x}{\frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\cos x}} \\~\\ =\frac{\cos x}{\frac{\sin^2 x+\cos^2 x}{\cos x}} \\~\\ =\frac{\cos x}{\frac{1}{\cos x}} \\~\\ = \cos^2 x$

:))