Question Number 191474 by Spillover last updated on 24/Apr/23
$$\mathrm{If}\:\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{cos}\:\mathrm{2A}+\mathrm{cos}\:\mathrm{2B}+\mathrm{cos2C}+\mathrm{1}=−\mathrm{4cosAcos}\:\mathrm{Bcos}\:\mathrm{C} \\ $$$$ \\ $$
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
![2cos(A+B)cos (A−B)+2cos^2 C−1+1 A+B+C=π A+B=π−C 2cos (π−C)cos (A−B)+2cos^2 C 2cos Ccos (A−B)+2cos^2 C 2cos C[cos (A−B)−cos C] C=π−(A+B) −2cos C[cos(A+B)+cos (A−B)] −2cos C[2cosAcos B] −4cos CcosAcos B](https://www.tinkutara.com/question/Q191545.png)
$$\mathrm{2cos}\left(\mathrm{A}+\mathrm{B}\right)\mathrm{cos}\:\left(\mathrm{A}−\mathrm{B}\right)+\mathrm{2cos}\:^{\mathrm{2}} \mathrm{C}−\mathrm{1}+\mathrm{1} \\ $$$$\:\:\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi\:\:\:\mathrm{A}+\mathrm{B}=\pi−\mathrm{C} \\ $$$$\mathrm{2cos}\:\left(\pi−\mathrm{C}\right)\mathrm{cos}\:\left(\mathrm{A}−\mathrm{B}\right)+\mathrm{2cos}\:^{\mathrm{2}} \mathrm{C} \\ $$$$\mathrm{2cos}\:\mathrm{Ccos}\:\left(\mathrm{A}−\mathrm{B}\right)+\mathrm{2cos}\:^{\mathrm{2}} \mathrm{C} \\ $$$$\mathrm{2cos}\:\mathrm{C}\left[\mathrm{cos}\:\left(\mathrm{A}−\mathrm{B}\right)−\mathrm{cos}\:\mathrm{C}\right] \\ $$$$\mathrm{C}=\pi−\left(\mathrm{A}+\mathrm{B}\right) \\ $$$$−\mathrm{2cos}\:\mathrm{C}\left[\mathrm{cos}\left(\mathrm{A}+\mathrm{B}\right)+\mathrm{cos}\:\left(\mathrm{A}−\mathrm{B}\right)\right] \\ $$$$−\mathrm{2cos}\:\mathrm{C}\left[\mathrm{2cosAcos}\:\mathrm{B}\right] \\ $$$$−\mathrm{4cos}\:\mathrm{CcosAcos}\:\mathrm{B} \\ $$$$ \\ $$