Question

please solve for 50 points

Note :- Don't spam otherwise I will reported your 10 answer ​

Answer 1

Hii,

Before you attempt to solve this question, it is worthy to know these formulas :-

$\frac{ {x}^{a} }{ {x}^{b} } = {x}^{a - b} \\ {x}^{a} \times {x}^{b} = {x}^{a + b} \\ {( {x}^{m}) }^{n} = {x}^{mn}$

Now, Coming to the question

$\frac{1}{1 + {x}^{a - b} } + \frac{1}{1 + {x}^{b - a} } \\ \\ = \frac{1}{1 + \frac{ {x}^{a} }{ {x}^{b} } } + \frac{1}{1 + \frac{ {x}^{b} }{ {x}^{a} } } \\ \\ = \frac{ {x}^{b} }{ {x}^{b} + {x}^{a} } + \frac{ {x}^{a} }{ {x}^{a} + {x}^{b} } \\ \\ = \frac{ {x}^{b} + {x}^{a} }{ {x}^{b} + {x}^{a} } \\ \\ = 1$

Hence proved .

Answer 2

Question:

$= > \frac{1}{1 + {x}^{a - b} } + \frac{1}{1 + {x}^{b - a} } = 1$

Answer:

$= > \frac{1}{1 + {x}^{a - b} } + \frac{1}{1 + {x}^{b - a} } = 1$

$= > \frac{1 + {x}^{b - a} + 1 + {x}^{a - b} }{1 + {x}^{b - a} + {x}^{a - b} + {x}^{a - b + b - a} } = 1$

$= > 2 + {x}^{b - a} + {x}^{a - b} = 1 + {x}^{b - a} + {x}^{a - b} + {x}^{0}$

$= > 2 + {x}^{b - a} + {x}^{a - b} = 1 + {x}^{b - a} + {x}^{a - b} + 1$

$= > 2 + {x}^{b - a} + {x}^{a - b} = 2 + {x}^{b - a} + {x}^{a - b}$

L.H.S. = R.H.S.

Hence, proved.

Hope this answer helps you ^_^ !