If $\mathrm{A}+\mathrm{B}+\mathrm{C}=\frac{3\mathrm{\pi }}{2}$,then cos2$\mathrm{A}+\cos 2\mathrm{B}+\cos 2\mathrm{C}$ is equal to

Since,$\mathrm{A}+\mathrm{B}+\mathrm{C}=\frac{3\mathrm{\pi }}{2}$

$\begin{array}{l}\therefore \cos 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{A}+\cos 2\mathrm{B}+\cos 2\mathrm{C}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=2\cos (\mathrm{A}+\mathrm{B})\cos (\mathrm{A}-\mathrm{B})+\cos 2\mathrm{C}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=2\cos \left(\frac{3\mathrm{\pi }}{2}-\mathrm{C}\right)\cos (\mathrm{A}-\mathrm{B})+1-2{\sin }^2\mathrm{C}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=1-2\sin \mathrm{C}\left[\cos (\mathrm{A}-\mathrm{B})+\sin \left(\frac{3\mathrm{\pi }}{2}-(\mathrm{A}+\mathrm{B})\right)\right]\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=1-2\sin \mathrm{C}[\cos (\mathrm{A}-\mathrm{B})-\cos (\mathrm{A}+\mathrm{B}\left)\right]\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=1-4\sin \mathrm{Asin}\mathrm{Bsin}\mathrm{C}\end{array}$