Question

$(a + \frac{1}{a} ) {2} - (a - \frac{1}{a} ) {2} = 4$
prove that​

Answer 1

To Prove:

$\large \bold{{(a + \frac{1}{a}) }^{2} - {(a - \frac{1}{a}) }^{2} = 4}$

Proof:

The given Equation can be proved if L.H.S = R.H.S

Now,

$L.H.S=$

$= {(a + \frac{1}{a}) }^{2} - {(a - \frac{1}{a} )}^{2}$

But,

We know that,

• ${x}^{2} - {y}^{2} = (x + y)(x - y)$

Therefore,

We get,

$= \large {(( a + \frac{1}{a} ) +( a - \frac{1}{a}) )((a + \frac{1}{a} )-( a - \frac{1}{a} ))} \\ \\ = \large{ (a + \frac{1}{a} + a - \frac{1}{a} )(a + \frac{1}{a} - a + \frac{1}{a} )} \\ \\ = \large{ (2a)( \frac{2}{a} )} \\ \\ = \large{4 \times \frac{a}{a} } \\ \\ = \large{4}$

$= R.H.S$

Hence,

L.H.S = R.H.S

Thus,

Proved