Question

prove that 1/1-x^m-n+1/1-x^n-m=1

Answer 1

L.H.S

1/[1-x^(m-n)] + 1/[1-x^(n-m)],

1/[1-x^m × x^(-n)] + 1/[1-x^n × x^(-m)],

x^n/(x^n - x^m) + x^m/(x^m - x^n),

x^n/(x^n - x^m) - x^m/(x^n - x^m),

now take the LCM here, we get

(x^n - x^m)/(x^n - x^m),

1

hence

L.H.S=R.H.S

Answer 2

Step-by-step explanation:

According to the question,

We need to prove LHS = RHS

So,

LHS

$1/[1-x^{m-n}] + 1/[1-x^{n-m}],\\\\1/[1-x^m X x^{-n}] + 1/[1-x^n X x^{-m}],\\\\x^n/(x^n - x^m) + x^m/(x^m - x^n),\\\\x^n/(x^n - x^m) - x^m/(x^n - x^m),$

now take the LCM here, we get

$x^n-x^m/x^n-x^m = 1$

LHS= 1

and From the question,

RHS also equal to 1

LHS=RHS

Hence, this equation proved.