The sum of cosines of direction angles \(\omega_i\) of a nonzero vector \(\mathbf{x}\) in \(\RNrSpc{n}\) satisfy
\[\cos^2 \omega_1\ +\ \cos^2 \omega_2\ +\ \cdots\ +\ \cos^2 \omega_n\ =\ 1\]The sum of cosines of direction angles \(\omega_i\) of a nonzero vector \(\mathbf{x}\) in \(\RNrSpc{n}\) satisfy
\[\cos^2 \omega_1\ +\ \cos^2 \omega_2\ +\ \cdots\ +\ \cos^2 \omega_n\ =\ 1\]We know that the \(i\)-th direction angle of \(\mathbf{x} = (x_1,\dots ,x_n)\) satisfies
\[\cos \omega_i = \dfrac{x_i}{ |\mathbf{x} | }\]Therefore,
\[\cos^2 \omega_i\ =\ \dfrac{ x_{i}^{2} }{ |\mathbf{x}|^2 }\ =\ \dfrac{ x_{i}^{2} }{ x_{1}^{2}+\cdots + x_{n}^{2} }\]Adding all of these terms up yields
\[\cos^{2} \omega_1 + \cdots + \cos^2 \omega_{n}^{2} = \dfrac{ x_{1}^{2}+\cdots + x_{n}^{2} }{ x_{1}^{2}+\cdots + x_{n}^{2} } = 1\]This was to be shown.