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\(\sin(x) + \cos(x) + \tan(x) + \cot(x) + \sec(x) + \csc(x) + 2 = 0\)

\(\begin{array}{clcc}& \sin(x) + \cos(x) + \tan(x) + \cot(x) + \sec(x) + \csc(x) + 2 & = & 0 \\&&& \\\Leftrightarrow & \sin(x) + \cos(x) + \dfrac{\sin(x)}{\cos(x)} + \dfrac{\cos(x)}{\sin(x)} + \dfrac{1}{\cos(x)} + \dfrac{1}{\sin(x)} + 2 & = & 0\end{array}\)

\(\begin{array}{clcc}\mbox{Bestaansvoorwaarde:} &&& \\&&& \\\cos(x) \neq 0 & \mbox{of} & \sin(x) \neq 0 & \\&&& \\\Leftrightarrow x \neq \pi + k\pi & \mbox{of} & x \neq k\pi \\\end{array}\)

\(\begin{array}{clcc}\Leftrightarrow & \sin^2(x)\cos(x) + \sin(x)\cos^2(x) + \sin^2(x) + \cos^2(x) + \cos(x) + \sin(x) + 2\sin(x)\cos(x) & = & 0 \\&&& \\\Leftrightarrow & \sin(x)\cos(x)(\sin(x) + \cos(x)) + (\sin(x) + \cos(x))^2 + \sin(x) + \cos(x) & = & 0 \\&&& \\\Leftrightarrow & (\sin(x) + \cos(x))(\sin(x)\cos(x) + \sin(x) + \cos(x) + 1) & = & 0 \\&&& \\\Leftrightarrow & (\sin(x) + \cos(x))(\sin(x)(\cos(x) + 1) + \cos(x) + 1) & = & 0 \\&&& \\\Leftrightarrow & (\sin(x) + \cos(x))(\sin(x) + 1)(\cos(x) + 1) & = & 0 \\\end{array}\)

\(\begin{array}{lclcc}\mbox{a)}\quad \sin(x) + \cos(x) = 0 && \mbox{b)}\quad \sin(x) + 1 = 0 && \mbox{c)}\quad \cos(x) + 1 = 0 \\&&&&& \\\Leftrightarrow \cos(x) = \cos\left(\dfrac{\pi}{2} + x\right) && \Leftrightarrow \sin(x) = -1 && \Leftrightarrow \cos(x) = -1 \\&&&&& \\\Leftrightarrow x = -\dfrac{\pi}{2} - x + 2k\pi && \Leftrightarrow x = \dfrac{3\pi}{2} + && \Leftrightarrow x = \pi + 2k\pi \\&&&&& \\\Leftrightarrow x = \dfrac{3\pi}{4} + k\pi && \Leftrightarrow x = \dfrac{3\pi}{2} + 2k\pi && \Leftrightarrow x = \pi + 2k\pi\end{array}\)

\(V = \left\{\frac{3\pi}{4} + k\pi\right\}\)