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

2secx 2cosx 2cosecx 2sinx

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