Cierra Castillo

2022-06-30

If $A+B+C+D=2\pi$, prove that:$\mathrm{cos}A+\mathrm{cos}B+\mathrm{cos}C+\mathrm{cos}D=4\mathrm{cos}\frac{A+B}{2}\cdot \mathrm{cos}\frac{A+C}{2}\cdot \mathrm{cos}\frac{B+C}{2}$

iskakanjulc

Expert

We will use $2\mathrm{cos}X\mathrm{cos}Y=\mathrm{cos}\left(X+Y\right)+\mathrm{cos}\left(X-Y\right)$
$\begin{array}{rl}4\mathrm{cos}\frac{A+B}{2}\mathrm{cos}\frac{A+C}{2}\mathrm{cos}\frac{B+C}{2}& =2\left(\mathrm{cos}\left(A+\frac{B+C}{2}\right)+\mathrm{cos}\frac{B-C}{2}\right)\mathrm{cos}\frac{B+C}{2}\\ & =2\mathrm{cos}\left(A+\frac{B+C}{2}\right)\mathrm{cos}\frac{B+C}{2}+2\mathrm{cos}\frac{B-C}{2}\mathrm{cos}\frac{B+C}{2}\\ & =\mathrm{cos}\left(A+B+C\right)+\mathrm{cos}A+\mathrm{cos}B+\mathrm{cos}C\\ & =\mathrm{cos}\left(2\pi -D\right)+\mathrm{cos}A+\mathrm{cos}B+\mathrm{cos}C\\ & =\mathrm{cos}A+\mathrm{cos}B+\mathrm{cos}C+\mathrm{cos}D\end{array}$

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