Simplify $\sec ^2 \alpha\left(1+\frac{1}{{cosec} \alpha}\right)\left(1-\frac{1}{cosec \alpha}\right)$.

1

sec²α ( 1 + \(\frac{1}{cosecα}\) ) . ( 1 - \(\frac{1}{cosecα}\) ) 

= sec²α (  \(\frac{cosecα + 1}{cosecα}\) ) . ( \(\frac{ cosecα -1}{cosecα}\) )
= sec²α (  \(\frac{cosec²α - 1²}{cosec²α}\) )

{ using , cosec²α - cot²α = 1 }

= sec²α (  \(\frac{cot²α}{cosec²α}\) )

= sec²α  × cos²α

= 1