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Simplify 2 + root 3 / 2 minus root 3 + 2 minus root 3 / 2 + root 3 + root3 - 1 / root 3 + 1

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\frac{2 + \sqrt{3} }{2 - \sqrt{3} } + \frac{2 - \sqrt{3} }{2 + \sqrt{3} } + \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1 } \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = \frac{2 + \sqrt{3} }{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } + \frac{2 - \sqrt{3} }{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } + \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1} \times \frac{ \sqrt{3} - 1 }{ \sqrt{3} - 1} \\ \\ using \: the \: identity \\ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{ {(2)}^{2} + {( \sqrt{3}) }^{2} + 2(2)( \sqrt{3} )}{ {(2)}^{2} - {( \sqrt{3} })^{2} } + \frac{ {(2)}^{2} + {( \sqrt{3}) }^{2} - 2(2)( \sqrt{3} )}{ {(2)}^{2} - {( \sqrt{3} })^{2} } + \frac{ {( \sqrt{3} )}^{2} + {(1)}^{2} - 2( \sqrt{3})(1) }{ {( \sqrt{3}) }^{2} - {(1)}^{2} } \\ \\ = \frac{4 + 3 + 4 \sqrt{3} }{4 - 3} + \frac{4 + 3 - 4 \sqrt{3} }{4 - 3} + \frac{3 + 1 - 2 \sqrt{3} }{3 - 1} \\ \\ = 7 + 4 \sqrt{3} + 7 - 4 \sqrt{3} + \frac{4 - 2 \sqrt{3} }{2} \\ \\ = 7 + 7 + 2 - \sqrt{3} \\ \\ = 16 - \sqrt{3}

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@Mahak24

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