Question

anyone help me please
$\frac{ \sqrt{x + 1} + \sqrt{x - 1} }{ (\sqrt{x + 1} - \sqrt{x - 1}) } + \frac{ \sqrt{x + 1} - \sqrt{x - 1} }{( \sqrt{x + 1} + \sqrt{x - 1}) }$
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Answer 1

$\\ \rm\hookrightarrow \dfrac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}+\dfrac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}$

$\sf Let\begin{cases}\sf \sqrt{x+1}=a \\ \sf \sqrt{x-1}=b\end{cases}$

Now

$\\ \rm\hookrightarrow \dfrac{a+b}{a-b}+\dfrac{a-b}{a+b}$

$\\ \rm\hookrightarrow \dfrac{\left\{(a+b)(a+b)\right\}+\left\{(a-b)(a-b)\right\}}{(a+b)(a-b)}$

$\\ \rm\hookrightarrow \dfrac{(a+b)^2+(a-b)^2}{a^2-b^2}$

$\\ \rm\hookrightarrow \dfrac{a^2+2ab+b^2+a^2-2ab+b^2}{a^2-b^2}$

$\\ \rm\hookrightarrow \dfrac{2a^2+2b^2}{a^2-b^2}$

$\\ \rm\hookrightarrow \dfrac{2(a^2+b^2)}{a^2-b^2}$

Put values

$\\ \rm\hookrightarrow \dfrac{2\left[(\sqrt{x+1})^2+(\sqrt{x-1})^2\right]}{(\sqrt{x+1})^2-(\sqrt{x-1})^2}$

$\\ \rm\hookrightarrow \dfrac{2(x+1)(x-1)}{(x+1)(x-1)}$

• Take 2 common

$\\ \rm\hookrightarrow 2\left(\dfrac{(x+1)(x-1)}{(x+1)(x-1)}\right)$

$\\ \rm\hookrightarrow 2(1)$

$\\ \rm\hookrightarrow 2$

Answer 2

Answer:

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