Prove that : cotA / (cosecA+1) + cosecA+1 / (cotA) = 2 secA
L.H.S.
= (cos A / sin A ) / (1/sin A + 1 ) + (1/sin A + 1 ) / (cos A / sin A )
= (cos A / sin A ) / ( 1+ sin A/ sin A) + ( 1+ sin A/ sin A) / (cos A / sin A )
= (cos A/ 1+ sin A) + (1+ sin A / cos A )
= cos2A + ( 1+ sin A)2 / cos ( 1+ sin A)
= cos2A + 1 + sin2A + 2 sin A / cos A( 1+ sin A)
= 1 + 1 + 2 sin A / cos A( 1+ sin A)
= 2 + 2 sin A / cos A( 1+ sin A)
= 2 ( 1+ sin A) / cos A( 1+ sin A)
= 2 / cos A
= 2 sec A = R.H.S.