[tex]\tan x+\frac{\cos x}{1+\sin x}\\\\ =\frac{\sin x}{\cos x}+\frac{\cos x}{1+\sin x}\\\\=\frac{\sin x(1+\sin x)+\cos^{2} x}{\cos x(1+\sin x)}\\\\=\frac{\sin^{2} x+\cos^{2} x+\sin x}{\cos x(1+\sin x)}\\\\=\frac{1+\sin x}{\cos x(1+\sin x)}\\\\=\frac{1}{\cos x}\\\\=\sec x[/tex]
