Question

Can anybody give the right answer i am too confused!​

Answer 1

Answer:

hey here is your answer

pls mark it as brainliest pls

Step-by-step explanation:

$so \: here \: \\ x = \frac{1}{2} ( \sqrt{a} + \frac{1}{ \sqrt{a} } ) \\ = \frac{1}{2} ( \frac{( \sqrt{a} ) {}^{2} + 1 }{ \sqrt{a} } ) \\ \\ = \frac{1}{2} ( \frac{a + 1}{ \sqrt{a} } ) \\ \\ ie \: \: \: x = \frac{a + 1}{2 \sqrt{a} }$

$so \: first \: finding \: value \: of \: \sqrt{x {}^{2} - 1} \\ \sqrt{x {}^{2} - 1} = \sqrt{( \frac{a + 1}{2 \sqrt{a} }) {}^{2} - 1 } \\ \\ = \sqrt{( \frac{a {}^{2} + 1 + 2a }{4a}) - 1 } \\ \\ = \sqrt{ \frac{a {}^{2} + 1 + 2a - 4a }{4a} } \\ \\ = \sqrt{ \frac{a {}^{2} - 2a + 1 }{4a} } \\ \\ = \sqrt{ \frac{(a - 1) {}^{2} }{(2 \sqrt{a} ) {}^{2} } } \\ \\ = \frac{a - 1}{2 \sqrt{a} }$

$so \: now \: substituting \: this \: value \: in \: to \: proven \: part \\ we \: get \\ \frac{ \sqrt{x {}^{2} - 1} }{x - \sqrt{x {}^{2} - 1} } \\ \\ = \frac{ \frac{a - 1}{2 \sqrt{a} } }{ \frac{a + 1}{2 \sqrt{a} } - \frac{(a - 1)}{2 \sqrt{a} } } \\ \\ = \frac{ \frac{ \frac{a - 1}{2 \sqrt{a} } }{a + 1 - (a - 1)} }{2 \sqrt{a} } \\ \\ = \frac{ \frac{ \frac{a - 1}{2 \sqrt{a} } }{a + 1 - a + 1} }{2 \sqrt{a} } \\ \\ = \frac{a - 1}{1 + 1} \\ \\ = \frac{a - 1}{2} \\ \\ hence \: lhs = rhs \\ thus \: proved$