Fatorando a expressão dada pela questão, temos que: $$ tg^2(x) + sen^2(x) = 3cos^2(x) \therefore tg^2(x) + [1-cos^2(x)] = 3cos^2(x) $$ $$ [tg^2(x) + 1] = 4cos^2(x) \therefore 4cos^2(x) = sec^2(x) \therefore cos^4(x) = \frac{1}{4} $$ Observando o resultado final, temos que as raízes serão dadas por: $$ cos^4(x) = \frac{1}{4} \therefore cos(x) = \pm \frac{√2}{2} $$ $$ \pu {x = \frac{\pi}{4}}; \pu{x = \frac{7\pi}{4}}; \pu{x = \frac{3\pi}{4}}; \pu{x = \frac{5\pi}{4}} $$ Portanto, a soma das raízes será $ \pu{4\pi} $ Gabarito Letra C!

Fatorando a expressão dada temos que : $\tan^2(x) + \sin^2(x) = 3\cos^2(x) = \dfrac{\sin^2(x)}{\cos^2(x)} + \sin^2(x)$ $\sin^2(x)+\sin^2(x)\cos^2(x) = 3\cos^4(x) = \sin^2(x) ( 1 + \cos^2(x))$ $ = 3\cos^4(x) = (1 - \cos^2(x))(1 + \cos^2(x)) $ $ = 3\cos^4(x) = 1 - \cos^4(x) \implies 4\cos^4(x) = 1 \implies \cos^4(x) = \dfrac{1}{4}$ $\implies \cos(x) = \pm \dfrac{\sqrt{2}}{2}$ $\implies \boxed{x = 45° = \dfrac{\pi}{4}}$ ou $\boxed{x = 135° = \dfrac{3\pi}{4}}$ ou $\boxed{x = 225° = \dfrac{5\pi}{4}} $ ou $\boxed{x = 315° = \dfrac{7\pi}{4}}$ Soma das raízes : $\dfrac{\pi}{4} + \dfrac{3\pi}{4} + \dfrac{5\pi}{4} + \dfrac{7\pi}{4}$ $ = \dfrac{16\pi}{4} = \boxed{4\pi}$ $\textbf{Resposta : Letra C}$