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Question
Evaluate the
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Solution
(ii) \[(x + \sqrt{x^2 - 1} )^6 + (x - \sqrt{x^2 - 1} )^6 \]
\[ = 2[ ^ {6}{}{C}_0 x^6 (\sqrt{x^2 - 1} )^0 +^{6}{}{C}_2 x^4 (\sqrt{x^2 - 1} )^2 +^{6}{}{C}_4 x^2 (\sqrt{x^2 - 1} )^4 + ^{6}{}{C}_6 x^0 (\sqrt{x^2 - 1} )^6 ]\]
\[ = 2[ x^6 + 15 x^4 ( x^2 - 1) + 15 x^2 ( x^2 - 1 )^2 + ( x^2 - 1 )^3 ]\]
\[ = 2[ x^6 + 15 x^6 - 15 x^4 + 15 x^2 ( x^4 - 2 x^2 + 1) + ( x^6 - 1 + 3 x^2 - 3 x^4 )]\]
\[ = 2[ x^6 + 15 x^6 - 15 x^4 + 15 x^6 - 30 x^4 + 15 x^2 + x^6 - 1 + 3 x^2 - 3 x^4 ]\]
\[ = 64 x^6 - 96 x^4 + 36 x^2 - 2\]
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