Question

Seja S a soma das raízes da equação cos x - tg x . sen x = 0 que pertencem ao intervalo [0,2π]. Determine S/4.

Answer 1

          $\cos x-\dfrac{\sin x}{\cos x}\cdot \sin x=0\\ \\ \cos x-\dfrac{\sin^2 x}{\cos x}=0\\ \\ \dfrac{\cos^2 x-\sin^2 x}{\cos x}=0\\ \\ \dfrac{\cos 2x}{\cos x}=0\\ \\ 2x \in \left\{\dfrac{\pi}{2}\;,\;\dfrac{3\pi}{2}\;,\;\dfrac{5\pi}{2}\;,\;\dfrac{7\pi}{2}\right\}\\ \\ x \in \left\{\dfrac{\pi}{4}\;,\;\dfrac{3\pi}{4}\;,\;\dfrac{5\pi}{4}\;,\;\dfrac{7\pi}{4}\right\}\\ \\ \\ S=\dfrac{\pi}{4}+\dfrac{3\pi}{4}+\dfrac{5\pi}{4}+\dfrac{7\pi}{4}\\ \\ \boxed{\boxed{S=4\pi}}$