cosx/(1+sinx)+(1+sinx)/cosx is equal to :

\(\frac{cosx }{1+sinx}\) + \(\frac{1+sinx }{cosx}\)

= \(\frac{cos²x + (1 + sinx )² }{cosx(1+sinx)}\)

= \(\frac{cos²x +sin²x  + 1 + 2sinx}{cosx(1+sinx)}\)

= \(\frac{2+ 2sinx}{cosx(1+sinx)}\)

= 2secx