Question

if tan theta + sin theta = M tan theta - sin theta = N prove that (mn) = 1

Answer 1

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