1+cscx - cotx = cosx
sec x
$\displaystyle \frac{1+cosec(x)}{sec(x)} = cos(x) + cot(x)$
$\displaystyle \frac{1+cosec(x)}{sec(x)} = cos(x)(1+\frac{1}{sin(x)})$
$\displaystyle 1+cosec(x)= sec(x)cos(x)(1+\frac{1}{sin(x)})$
$\displaystyle 1+cosec(x)= \frac{1}{cos(x)}cos(x)(1+cosec(x))$
$\displaystyle 1+cosec(x)= 1+cosec(x)$
QED!