Recall the definitions of tangent and cotangent:
[tex]\tan x=\dfrac{\sin x}{\cos x}[/tex]
[tex]\cot x=\dfrac1{\tan x}=\dfrac{\cos x}{\sin x}[/tex]
Then
[tex]\sin x(\cot x+\tan x)=\sin x\left(\dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}\right)=\cos x+\dfrac{\sin^2x}{\cos x}[/tex]
Recall that [tex]\sin^2x+\cos^2x=1[/tex]:
[tex]\sin x(\cot x+\tan x)=\cos x+\dfrac{1-\cos^2x}{\cos x}=\cos x+\dfrac1{\cos x}-\cos x=\dfrac1{\cos x}[/tex]
and the definition of secant,
[tex]\sec x=\dfrac1{\cos x}[/tex]
