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if tan theta +sin theta=m, tan theta -sin theta=n show that m square -n square=4 root mn

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Given :- (1)  TanA + SinA = m

  (2)  TanA - SinA = n

To prove :- m2 - n2 = 4  x  (root mn)

 

Proof :-

L.H.S =  m2 - n2

  = (TanA + SinA)2 - (TanA - SinA)2

  = (Tan2A + Sin2A + 2 TanA SinA) - ( Tan2A + Sin2A - 2 TanA SinA)

  = Tan2A + Sin2A + 2 TanA SinA - Tan2A - Sin2A + 2 TanA SinA

  = Tan2A + Sin2A + 2 TanA SinA - Tan2A  - Sin2A + 2 TanA SinA

  = 4 TanA SinA

R..H.S ;- 4 root mn

  = 4 x [ root (TanA + SinA)(TanA - SinA) ]

  = 4 x [ root ( Tan2A - Sin2A ) ]

  = 4 x [ root ( Sin2A / Cos2A  -  Sin2A )

  = 4 x [ root { (Sin2A - Sin2ACos2A) / Cos2A } ]

  = 4 x [ root { Sin2A (1 - Cos2A) } / Cos2A ]

  = 4 x [ root { (Sin2A  x  Sin2A ) / Cos2A ]

  = 4 x (root [ Sin2A Tan2A ] )

  = 4 TanA SinA = L.H.S


Hence Proved

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