Question Number 219659 by Nicholas666 last updated on 30/Apr/25
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}; \\ $$$$\:\:{cos}\:\left({B}+{C}−{A}\right)−{cos}\left({C}+{A}−{B}\right)+{cos}\left({A}+{B}−{C}\right)−{cos}\left({A}+{B}+{C}\right)\:=\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}{sinAcosBsinC} \\ $$$$ \\ $$
Answered by som(math1967) last updated on 30/Apr/25
$${cos}\left({B}+{C}−{A}\right)−{cos}\left({C}+{A}−{B}\right) \\ $$$${cos}\left({A}+{B}−{C}\right)−{cos}\left({A}+{B}+{C}\right) \\ $$$$=\mathrm{2sin}\:\frac{\mathrm{2}{C}}{\mathrm{2}}\mathrm{sin}\:\left(\frac{\mathrm{2}{A}−\mathrm{2}{B}}{\mathrm{2}}\right)+\mathrm{2sin}\:\frac{\mathrm{2}\left({A}+{B}\right)}{\mathrm{2}}\mathrm{sin}\:\left(\frac{\mathrm{2}{C}}{\mathrm{2}}\right) \\ $$$$=\mathrm{2sin}\:{C}\left\{\mathrm{sin}\:\left({A}−{B}\right)+\mathrm{sin}\:\left({A}+{B}\right)\right\} \\ $$$$=\mathrm{2sin}\:{C}×\mathrm{2sin}\:\frac{\mathrm{2}{A}}{\mathrm{2}}×\mathrm{cos}\left(\:\frac{\mathrm{2}{B}}{\mathrm{2}}\right) \\ $$$$=\mathrm{4sin}\:{A}\mathrm{cos}\:{B}\mathrm{sin}\:{C} \\ $$