$\frac{cotx}{1+cosecx}+\frac{1+cosecx}{cotx}$ is equal to :

2secx

\(\frac{cotx}{1+cosecx}\) + \(\frac{1+cosecx }{cotx}\)

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

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

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

{ sin²x + cos²x = 1 }

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

= 2secx