The question pertains to Trigonometric element in Determinants.
I tried but could not get RHS.
2026-03-31 19:10:40.1774984240
Determinants Proof Problem
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$$D=\sin(C-B)+\sin(A-C)+\sin(B-A)$$
Set $C-B=2X$ etc. $X+Y+Z=0,X+Y=-Z,$
$\cos(X+Y)=\cos(-Z)=?$
$\sin(X+Y)=\sin(-Z)=?$
$$=\sin2X+\sin2Y+\sin2Z$$
$$=2\sin(X+Y)\cos(X-Y)+2\sin Z\cos Z$$
$$=-2\sin Z\cos(X-Y)+2\sin Z\cos(X+Y)$$
$$=-2\sin Z[\cos(X-Y)-\cos(X+Y)]$$
$$=-2\sin Z[2\sin X\sin Y]$$