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Question

Given x=a2+b2+a2b2a2+b2a2b2
use componendo and dividend0 to prove that :
b2=2a2xx2+1.

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Solution

x=a2+b2+a2b2a2+b2a2b2
Applying Componendo and Dividendo
x+1x1=a2+b2+a2b2+a2+b2a2b2a2+b2+a2b2a2+b2+a2b2
x+1x1=a2+b2a2b2
Squaring both sides
(x+1x1)2=a2+b2a2b2
Applying Componendo & Dividendo
(x+1)2+(x1)2(x+1)2(x1)2=a2b2
2(x2+1)2(2x)=a2b2
x2+12x=a2b2
b2=2a2xx2+1 (RHS)
(Proved)

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