Question

cosec A - 1/ cosec A + 1 = ( cos A / 1 + sin A)^2

Answer 1

cosec A - 1 = (1/sinA) -1
= (1-sinA) / sinA

cosec A + 1 = (1/sinA) +1
= (1+sinA) / sinA

Now, (cosec A - 1) / (cosec A + 1) =
[ (1-sinA) / sinA ] / [ (1+sinA) / sinA ]
= [ (1-sinA) / sinA ] x [ sinA / (1+sinA) ]
= 1-sinA / 1+sinA
Now multiplying numerator and denominator by 1+sinA we get
= ( 1-sinA )x( 1+sinA ) / ( 1+sinA )²
= (1-sin²A) / ( 1+sinA )²

But ,
cos²A+ sin²A =1
cos²A = 1 - sin²A
So now substituting this value of 1 - sin²A
cosec A - 1 / cosec A + 1 = cos²A / ( 1+sinA )²
=RHS

Hence proved

Answer 2

LHS
(cosec A -1)(cosec A +1)
————————————
(cosec A +1)(cosec A +1)

= cosec²A - 1
——————
(cosec²A + 1)²

= cot²A
————————————
cosec²A +1 +2 cosecA

=. 1
——————————————
sec²A + tan²A +2secA tanA. ( dividing both side by cot²A)

= cos²A
——————————————. ( multiplying both sides by cos²A)
1 + sin²A +2sinA

=. (cosA)²
———————
(1 + sin A)²

=. RHS


Hence proved

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