Simplifique a expressão:

Definindo,

[latex]S = \frac{\log {\sin^4 {x}}}{\log {\left (1+\cos {x} \right )} + \log {\left (1-\cos {x} \right )}}[/latex]

Temos:

[latex]S = \frac{\log {\sin^4 {x}}}{\log {\left (2 \cos^2 {\frac{x}{2}} \right )} + \log {\left (2\sin^2 {\frac{x}{2}} \right )}} = \frac{\log {\sin^4 {x}}}{\log {\left (4 \cos^2 {\frac{x}{2} \sin^2 {\frac{x}{2}}} \right )}}[/latex]

[latex]S = \frac{\log {\sin^4 {x}}}{\log {\left (4 \cos^2 {\frac{x}{2} \sin^2 {\frac{x}{2}}} \right )}} = \frac{\log {\sin^4 {x}}}{\log {\sin^2 {x}}}[/latex]

[latex]S = \frac{\log {\sin^4 {x}}}{\log {\sin^2 {x}}} = \frac{4\cdot \log {\sin {x}}}{2\cdot \log {\sin {x}}} = 2[/latex]

Com isso,

[latex]S = 2[/latex]