cot^2A/1+cosecA =1/sinA

1.	cot^2A/1+cosecA =1/sinA
Answer» As we know1/sin a= cosec a LHS = 1 + (Cot²A) / (1+CosecA) = {1 +CosecA + Cot²A} / ( 1+ CosecA) On multiplying Numerator & denominator by ( (1- CosecA) = {( 1+CosecA +Cot²A) ( 1-CosecA)} / (1-Cosec²A) = (1+CosecA+Cot²A-CosecA -Cosec²A -Cot²A.CosecA) / (1-Cosec²A) Using fundamental trigonometric identity: Cot²A+1 = Cosec²A , DENOMINATOR = 1-Cosec²A= -Cot²A And NUMERATOR = 1+Cot²A-Cosec²A-Cot²A.CosecA = Cosec²A-Cosec²A - Cot²A.CosecA = -Cot²A.CosecA Now, LHS = (-Cot²A.CosecA)/(-Cot²A) = CosecA= RHS [Hence Proved]

Answer»

As we know1/sin a= cosec a

LHS = 1 + (Cot²A) / (1+CosecA)

= {1 +CosecA + Cot²A} / ( 1+ CosecA)

On multiplying Numerator & denominator by ( (1- CosecA)

= {( 1+CosecA +Cot²A) ( 1-CosecA)} / (1-Cosec²A)

= (1+CosecA+Cot²A-CosecA -Cosec²A -Cot²A.CosecA) / (1-Cosec²A)

Using fundamental trigonometric identity:

Cot²A+1 = Cosec²A , DENOMINATOR = 1-Cosec²A= -Cot²A

And NUMERATOR = 1+Cot²A-Cosec²A-Cot²A.CosecA

= Cosec²A-Cosec²A - Cot²A.CosecA

= -Cot²A.CosecA

Now, LHS = (-Cot²A.CosecA)/(-Cot²A)

= CosecA= RHS

[Hence Proved]

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