Sin²α(1+sin^-1α+ ctgα)(1-sin^-1α+ctgα)

Sin²α·(1+1/sinα + ctgα)(1-1/sinα+ctgα) =

= Sin²α·(1+ ctgα +1/sinα )(1 +ctgα-1/sin²α) =

= Sin²α·[(1 +ctgα)²-1/sin²α] =

= Sin²α·[(1 +ctgα)²·sin²α - 1]/sin²α =

= (1 +ctgα)²/(1 + ctg²α) - 1 =

= (1 + ctg²α + 2ctgα)/(1 + ctg²α) - 1 =

= 1 + 2ctgα/(1 + ctg²α) -1

= 2ctgα/(1 + ctg²α) =

= 2ctgα/(1 + ctg²α) =

= 2ctgα·sin²α =

= 2cosα·sin²α /sinα =

= 2cosα·sinα =

= sin2α