Question

prove the following

(cotA/cosecA+1)+(cosecA+1/cotA)=2secA​

Answer 1

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.

L.H.S = R.H.S.

Hence, Proved

Answer 2

cotA/(cosecA+1) + (cosecA+1)/cotA

=cot²A+(cosecA+1)²/cotA(cosecA+1)

=(cot²A+cosec²A+1+2cosecA)/cotA(cosecA+1)

=(2cosec²A+2cosecA)/cotA(cosecA+1)

=2cosecA(cosecA+1)/cotA(cosecA+1)

=2cosecA/cotA

=2secA