Question

( cotA+cosecA-1 )/( cotA-cosecA+1 ) = ( 1+cosA )/sinA

Answer 1

LHS = Cot A + Cosec A - 1 / Cot A - Cosec A + 1

=> ( Cosec A + Cot A ) - ( Cosec²A - Cot²A ) / ( Cot A - Cosec A + 1 ). [ Since 1 = Cosec² theta - Cot² theta ]

=> ( Cosec A + Cot A ) { 1 - ( Cosec A - Cot A )} / ( Cot A - Cosec A + 1 ) .

=> ( Cosec A + Cot A ) ( Cot A - Cosec A + 1 ) / ( Cot A - Cosec A + 1 ).

=> ( Cosec A + Cot A )

=> ( 1/ Sin A + CosA/SinA )

=> ( 1 + CosA ) / ( Sin A ) = RHS.

Hence,

LHS = RHS = ( 1 + Cot A / SinA ).