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Question Number 191474 by Spillover last updated on 24/Apr/23

$$\mathrm{If}\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$$$$\mathrm{Prove}\mathrm{that}$$$$\cos 2\mathrm{A}+\cos 2\mathrm{B}+\cos 2\mathrm{C}+1=-4\mathrm{cosAcos}\mathrm{Bcos}\mathrm{C}$$$$$$

Commented by Tinku Tara last updated on 24/Apr/23

$$\mathrm{You}\mathrm{probably}\mathrm{need}\mathrm{precondition}$$$$\mathrm{that}\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$$

Answered by Spillover last updated on 25/Apr/23

$$2\cos (\mathrm{A}+\mathrm{B})\cos (\mathrm{A}-\mathrm{B})+2\cos {}^2\mathrm{C}-1+1$$$$\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi \mathrm{A}+\mathrm{B}=\pi -\mathrm{C}$$$$2\cos (\pi -\mathrm{C})\cos (\mathrm{A}-\mathrm{B})+2\cos {}^2\mathrm{C}$$$$2\cos \mathrm{Ccos}(\mathrm{A}-\mathrm{B})+2\cos {}^2\mathrm{C}$$$$2\cos \mathrm{C}[\cos (\mathrm{A}-\mathrm{B})-\cos \mathrm{C}]$$$$\mathrm{C}=\pi -(\mathrm{A}+\mathrm{B})$$$$-2\cos \mathrm{C}[\cos (\mathrm{A}+\mathrm{B})+\cos (\mathrm{A}-\mathrm{B})]$$$$-2\cos \mathrm{C}\left[2\mathrm{cosAcos}\mathrm{B}\right]$$$$-4\cos \mathrm{CcosAcos}\mathrm{B}$$$$$$

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