We have the following:
[tex]\cos x+\tan x\cdot\sin x[/tex]we simplify:
[tex]\cos x+\frac{\sin x}{\cos x}\cdot\sin x=\cos x+\frac{\sin^2x}{\cos x}=\frac{\cos x\cdot\cos x}{\cos x}+\frac{\sin^2x}{\cos x}=\frac{\cos^2x}{\cos x}+\frac{\sin^2x}{\cos x}=\frac{\cos^2x+\sin^2x}{\cos x}=\frac{1}{\cos x}=\sec x[/tex]Therefore
[tex]\cos x+\tan x\cdot\sin x=\sec x[/tex]