Sin(x)/Cos(x)

I assume that you really mean this     

Sin(x)/Cos(x) + Cos(x)/(1 + Sin(x) )

$$\\\frac{Sin(x)}{Cos(x)} + \frac{Cos(x)}{1 + Sin(x) }\\\\
=\frac{Sin(x)}{Cos(x)} + \frac{Cos(x)(1-sin(x)) }{(1 + Sin(x))(1-sin(x)) }\\\\
=\frac{Sin(x)}{Cos(x)} + \frac{Cos(x)(1-sin(x)) }{(1-sin^2(x)) }\\\\
=\frac{Sin(x)}{Cos(x)} + \frac{Cos(x)(1-sin(x)) }{cos^2(x) }\\\\
=\frac{1}{Cos(x)}\\\\
=sec(x)$$