Question

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Prove that :-
cos A cot A / 1- sin A = 1+ cosec A

Answer 1

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Given that, cosA×cotA/1-sinA = 1+ cosecA


=)Consider L.H.S we get

=) cosA.cotA/1-sinA

=)cosA.cosA/sinA whole divided by 1-sinA

=)cos2A/sinA whole divided bu 1-sinA

=) 1-sin2A/sinA×1-sinA

=) 1-sin2A×1+sinA/sinA.( 1-sinA).(1+sinA)

=)(1-sin2A)× (1+sinA)/sinA. (1-sin2A)

=) 1+sinA/sinA

=) [ 1/sinA]+1

=) 1+ cosec A =R.H.S

Hence, proved..


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Answer 2

Given :  CosA * cotA/ ( 1 + SinA)  = Cosec A - 1

To Find : Prove

Solution:

CosA * cotA/ ( 1 + SinA)  = Cosec A - 1

LHS = CosA * cotA/ ( 1 + SinA)

= Cos A. Cos A/(SinA )(1 + SinA)

= Cos²A/ SinA ( 1 + SinA)

= ( 1 - Sin²A) /SinA(1 + SinA)

= (1 + SinA)(1 - SinA)/(SinA.(1 + SinA))

=( 1  - SinA)/SinA

= 1/SinA  - SinA/SinA

= CosecA - 1

= RHS

QED

Hence proved

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