Log of Sine

${ e }^{ (\sin ^{ 2 }{ x } +\sin ^{ 4 }{ x } +\sin ^{ 6 }{ x } +..........\infty )\ln { 2 } }$

$\rightarrow (sin ^{ 2 }{ x } +\sin ^{ 4 }{ x } +\sin ^{ 6 }{ x } +..........\infty )\ln { 2 } = tan^{2}{x}\ln{ 2 }$ by using sum of infinite series in a geometrical progression, $\frac{a}{1-r}$ if $|r| < 1$

$a = r = sin^{2}{x}$

Hence $2^{tan^{2}{x}}$ should satisfy the quadratic equation $x^2 - 9x + 8$

On solving , we get $tan{x} = 0$ & $tan^{2}{x} = 3$ which gives $x = 0 , 60^{0} , 120^{0}$

$g(x) = \frac{ cos{x} } { cos{x} + sin{x} } = \frac{\sqrt{3} - 1}{2} , \frac{\sqrt{3} + 1}{2}$