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Solving a Quadratic Equation by Factorization Method

If cosec p-si...

Question

If (cosecp-sinp) = a³and (secp-cosp) = b³
Prove that a²b²(a²+b²)=1

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Solution

Given,
cosec P- sin P = a³ & sec P - cosP = b³
To prove : a²b² ( a² + b² ) = 1
1/ sinP - sinP = a³
or ( 1 - sin²P)/sinP = a³
or cos²P/sinP = a³
or cotP .cosP = a³-------(I)
secP - cos P =b³
or sin²P/cosP = b³
or tanP sinP = b³------(II)
cot P.cosP/tanP sinP = a³/b³
or cot³P = a³/b³
or cot P = a/b
or a = b cotP
L . H. S. = a²b²( a² + b²)
= b²cot²P. b²( cot²P b² + b²)
= b⁶ cot²P( cot²P +1)
= tan²P sin²P .cot²P. cosec²P
= (tan²P/tan²P)* (sin²P/sin²P)
= 1 = R. H. S. Q.E.D.

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