Question

plz solve both of them

Answer 1

According to given sum,

(i)

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

$= > \: \frac{1}{a} ( \frac{1}{ {a}^{ - 1} + {b}^{ - 1} } ) + \frac{1}{a}( \frac{1}{ {a}^{ - 1} - {b}^{ - 1} } )$

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

$= > \: \frac{b}{a + b} + \frac{b}{b - a}$

$= > \: b( \frac{1}{a + b} + \frac{1}{b - a} )$

$= > \: b( \frac{b - a + b + a}{ {b}^{2} - {a}^{2} } )$

$= > \: b( \frac{2b}{ {b}^{2} - {a}^{2} } )$

$= > \: \frac{2 {b}^{2} }{ {b}^{2} - {a}^{2} }$

LHS = RHS .

Hence, proved.

(ii)

$lhs = \frac{ {a} + {b} + {c} }{ {a}^{ - 1}. {b}^{ - 1} + {b}^{ - 1}. {c}^{ - 1} + {a}^{ - 1} . {c}^{ - 1} }$

$= > \: \frac{a + b + c}{ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} }$

Taking the LCM of the denominators.

$= > \: \frac{a + b + c}{ \frac{a + b + c}{abc} }$

$= > \: abc( \frac{a + b + c}{a + b + c} )$

=> abc.

LHS = RHS .

Hence, proved

:-)Hope it helps u .