Question

Total number of values of $x$ satisfying $(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=2^{3x}$ is

Answer 1

$(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=2^{3x}$
The above equation can be re-written as
$(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=(2\sqrt{2}{)}^{2x}$
$\Rightarrow {\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)}^{2x}+{\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^{2x}=1$
$\Rightarrow (\sin {75}^{\circ }{)}^{2x}+(\cos {75}^{\circ }{)}^{2x}=1$
$\therefore x=1$