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