Consider the LHS.
⇒cotA+cscA−1cotA−cscA+1
We know that,
csc2A−cot2A=1
a2−b2=(a−b)(a+b)
Therefore,
⇒cotA+cscA−(csc2A−cot2)cotA−cscA+1
⇒cotA+cscA[1−cscA+cotA]cotA−cscA+1
⇒cotA+cscA
⇒cosAsinA+1sinA
⇒1+cosAsinA