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Question

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

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Solution

x=a2+b2+a2b2a2+b2a2b2

Use componendo and divivdendo rule:

x+1x1=a2+b2+a2b2+a2+b2a2b2a2+b2+a2b2a2+b2+a2b2

x+1x1=2a2+b22a2b2

squaring on both sides

(x+1)2(x1)2=2(a2+b2)2(a2b2)

(x+1)2(x1)2=(a2+b2)(a2b2)

(a2b2)(x2+1+2x)=(a2+b2)(x2+12x)

4a2x2b2x22b2=0

2a2xb2(x2+1)=0

b2=2a2xx2+1

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