Reciprocal trig, addition formulae, double angle formula, product to sum, sum to product formulae, harmonic formulae, inverse functions
Show that if at least one of the four angles \(A\pm B\pm C\) is a multiple of \(\pi\), then \begin{alignat*}{1} \sin^{4}A+\sin^{4}B+\sin^{4}C & -2\sin^{2}B\sin^{2}C-2\sin^{2}C\sin^{2}A\\ & -2\sin^{2}A\sin^{2}B+4\sin^{2}A\sin^{2}B\sin^{2}C=0. \end{alignat*}