Official solution
Let \[{\left( {a + x} \right)^n}\] = Odd trems(A) + Even terms(B)
So \[{\left( {a - x} \right)^n}\] = Odd terms(A) - Even terms(B)
\[\therefore\] \[{\left( {a + x} \right)^n} - {\left( {a - x} \right)^n}\]
= (A + B) + (A - B)
= 2A
= 2[odd terms]
= 2[ T1 + T3 + T5 + ....... ]
So in case of
\[{\left( {x + \sqrt {{x^3} - 1} } \right)^6}\] + \[{\left( {x - \sqrt {{x^3} - 1} } \right)^6}\]
= 2[ T1 + T3 + T5 + T5 ]
= 2[ \[{}^6{C_0}{x^6} + {}^6{C_2}{x^4}\left( {{x^3} - 1} \right)\]
\[ + {}^6{C_4}{x^2}{\left( {{x^3} - 1} \right)^2} + {}^6{C_6}{\left( {{x^3} - 1} \right)^3}\]]
= 2[ \[{x^6} + 15{x^4}\left( {{x^3} - 1} \right) + 15{x^2}{\left( {{x^3} - 1} \right)^2} + {\left( {{x^3} - 1} \right)^3}\] ]
= 2[ \[{x^6} + 15\left( {{x^7} - {x^4}} \right)\]
\[ + 15{x^2}\left( {{x^6} - 2{x^3} + 1} \right) + \left( {{x^9} - 3{x^6} + 3{x^3} - 1} \right)\] ]
\[ \therefore \] Sum of coefficient of all even degree terms
= 2[ 1 - 15 + 15 + 15 - 3 - 1 ] = 24