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If A, B, C, D be the angles of acyclic q...

If `A, B, C, D` be the angles of acyclic quadrilateral, show that : `cosA +cosB+cosC+cosD=0`.

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We know that the opposite angles of a cyclic quadrilateral are supplementary, i.e., `A+C=pi" and " B+ D=pi`. Therefore,
`A=pi-C" and " B=pi-D`
`rArr cosA=cos(pi-C)=-cosD`
and `cosB=cos(pi-D)=-cosD`
`:. cosA+cosB+cosC+cos D`
`=-cosC-cosD+cosC+cosD=0`
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