Q. 184.7( 3 Votes )

Prove that:
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Answer :


Put x = cos2θ


We know that cos2θ = 1 – 2sin2θ and cos2θ = 2cos2θ – 1


Hence 1 + cos2θ = 2cos2θ and 1 – cos2θ = 2sin2θ


1 + x = 1 + cos2θ = 2cos2θ


1 – x = 1 – cos2θ = 2sin2θ


Consider LHS




Cancel out √2





We know that



Using




As x = cos2θ cos-1x = 2θ θ = 1/2(cos-1x)



LHS = RHS


Hence proved


OR



We know that


Here and




As




x2 + 4x – 2x – 8 + x2 – 4x + 2x – 8 = x2 – 16 – (x2 – 4)


2x2 – 16 = x2 – 16 – x2 + 4


2x2 = 4


x2 = 2


x = ±√2


Hence x is ±√2.

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