Question

Prove that
a^-1/a^-1+b^-1 +a^-1/a^-1-b^-1=2b^2/b^2-a^2

Answer 1

To prove that a-1 /a-1 +b-1 +a-1 /a-1 -b-1 = 2b2 /b2 -a2Lets take the LHS (Left Hand Side) first:

=1/a÷1/a+1/b + 1/a÷1/a -1/b

Next take the LCM for the denominator, so it becomes:

=1/a÷b+a /ab + 1/a÷b-a /ab

=1/a×ab /b+a+ 1/a×ab /b-a

Now 'a' gets cancelled in both the numerator and the denominator. So it becomes:

=1/a×ab /b+a+ 1/a×ab /b-a

=b /b+a+b /b-a

take the LCM of the denominator and so it becomes:

=b(b-a)/(b+a)(b-a)+b(b+a)/(b-a)(b+a)

=b2 -ab /b2 -a2 +b2 +ab /b2 -a2

=b2 -ab +b2 +ab /b2 -a2

=b2+b2/b2-a2

=2b2/b2 -a2 ==== RHS (Right Hand Side)