Question

Evaluate $[x+\sqrt{(x^2-1)}{]}^6+[x-\sqrt{(x^2-1)}{]}^6$

Answer 1

As above $X=x,a=\sqrt{X^2-1}$
$\therefore$ ${\left[(X+\sqrt{X^2-1})\right]}^6+{\left[(X-\sqrt{X^2-1})\right]}^6$
$=2[X^6+15X^4\sqrt{(X^2-1{)}^2}+15X^2\sqrt{(X^2-1{)}^4)}+\sqrt{X^2-1{)}^6}]$
$=2[X^6+15X^4(X^2-1)+15X^2(X^2-1{)}^2+(X^2-1{)}^3]$

$=2[X^6+15(X^6-X^4)+15(X^6-2X^4+X^2)+(X^6-3X^4+3X^2-1\left)\right]$

$64X^6-96X^4+36X^2-2$